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Relative Velocity

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Published in: Mathematics
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A comprehensive document that discusses relative velocity with theory and lots of practical examples.

Maged / Abu Dhabi

15 years of teaching experience

Qualification: B.Sc. in Mathematics

Teaches: Mental Maths, Vedic Maths, Statistics, Maths, Advanced Maths, Mathematics, GRE, SAT, ACT, EMSAT, IB Exam Preparation, Exam Preparation, Computer

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  1. Relative Velocity Overview Category 1: Two Independent Bodies. Introduction (i) General Questions (ii) Interception (iii) Shortest Distance (iv) Distance from Intersection (v) Distance from each other Category 2: Body and Carrier Cate o 3: Wind A Introduction Exam questions ars Introduction Exam Questions Answers to Ordinary Level Exam Questions Guide to answering Higher Level Exam Questions 2009 - 1996 Page 2 2 4 7 9 14 16 19 21 24 25 27 29 Higher Level Marking Schemes to be provided separately Questions to make you think . You are travelling in a car. You have a glass bottle in your hand. In which direction should you throw it to minimise the danger of its breaking on hitting the ground? 2. Two friends are playing with a ball on board a ship moving at a steady speed. One is standing nearer the aft and the other nearer the bows. Which one of them finds it easier to throw the ball to the other? (Ignore wave and wind effects)
  2. Overview Relative Velocity questions can be sub-divided into three categories: l. Two Independent Bodies 2. Body and Carrier 3. Wind Appears Category 1: Two Independent Bodies Introduction For these questions we will always have two bodies (usually cars or ships) moving independently Of each other. You're in a car (let's call it car B, because maths people are seriously devoid of imaginations — why couldn't they call it "Car BG" —for Bad Guys?) travelling along a road at 20 m s-l. Another car (car A) is following behind you travelling at 25 m s-l. It's obvious (isn't it?) that the car is gaining on you as if you were stopped and car A was travelling towards you at 5 m s-l We can say that the velocity of car B relative to car A is 5 m s-l. We write this mathematically as follows: Vab = 5 m s-l We can always work out the velocity Of A relative to B by using the formula Vab = Va — Vb This works regardless of which car is quicker, but note that the signs are important. For instance if car A was travelling at 25 m s-l and car B was following at a velocity of 20 m s-l then Vab = - 5 m s-l, i.e. the velocity of A relative to B is minus 5 m s-l, which means that as car B (you) see it, car A is moving backwards at a speed of 5 m But if you were in car A what would the scenario look like? Well now we need to find Vba; the velocity of car B relative to car A. Vba = Vb— Va = 25 —20= 5 m s-l, i.e. the occupants Of car A see car B moving ahead of them at a speed of 5m s l. One more look at this If car A, travelling at 120 kilometres per hour collides with your car (car B) coming in the opposite direction but also travelling at 120 kilometres per hour, the velocity of car A relative to you is 240 kilometres per hour (work it Out, and while you're at it work Out the velocity Of your car relative to Car A), i.e. it is as if you were stopped and a car ran into you while travelling at that velocity. Maybe if the human brain had evolved to think in terms of relative speeds it might make us pay a little more attention when we drive in what are essentially death machines. Or maybe not Time One final point to note before we move on is the following: Let's go back to the situation we had at the beginning - you're in car B travelling along a road at 20 m s-l while car A is following behind you travelling at 25 m s- . As before, Vab = 5 m s-l. Now let's assume that at a given moment in time car A is 100 m behind you. How long will it take car A to catch up with car B? Answer: Time = distance/velocity = 100/5 = 20 seconds. Notice that this works even though we are using relative velocities, not actual velocities. 2
  3. Okay. So far so good. Now where all this relative velocity malarkey starts to really pay its way is when the two objects are moving at an angle to each other (relative to a fixed axis like a road junction for cars or compass points for ships or planes). You will have noticed by now that text-books and exam papers tend to use the word 'bodies' a lot. This is because in olden days life was cheap and all experiments involving moving objects were as easily done with human bodies as they were with expensive machinery — unfortunately we're not allowed fire actually bodies Out Of cannons and such anymore, but at least we'll this terminology we always have the memories. If we want to find out how close two bodies come to each other in subsequent motion then the only way to figure it out is by using relative velocity. In this case there is an i and a j component so care must be taken when working out Vab. e.g. if Va = 2 i + 3 j and Vb = i + 2j then V ab = (2 — l) i + (3 — 2) j = i + j Notice that Vba = - i —j From this any one of a number of questions could be asked: (i) General questions (ii) Time for interception to occur (iii)Shortest distance in subsequent motion (iv)Distance each object is from the intersection when the objects are nearest to each other (v) Calculate the length of time for which the objects are less than or equal to 9 km apart. We will look at each of these types of question separately. 3
  4. Category 1: Two Independent Bodies (i) General Questions 2000 (a) OL Ship A is travelling with a speed of 15 km/hr in the direction due East. Ship B is travelling with a speed of 20 km/hr in the direction due South. Find the velocity of ship A relative to ship B. Solution Val) = Va-Vb = (15 i) - 20 j) = 15i+20j 2004 (b) OL Car P and car Q are travelling eastwards on a straight level road. P has a constant speed of 20 m/s and Q has a constant speed of 10 m/s. (i) Find the velocity of P relative to Q. (ii) At a certain instant car P is 100 m behind car Q. Find the distance between the two cars 3.5 seconds later. Solution (i) vpq=vp-vq = (20 i) - (10 i) =10i+Oj (ii) Distance = 100 + Sq — Sp = 100 + = 65 m 2010 OL A river is 100 metres wide and has parallel banks. Boat B departs from point P on its western bank and lands at point Q on its eastern bank. The actual velocity of the boat is 5 i + 12 j m s- Cyclist C travels due north at a constant speed of 3 m s-l along the eastern bank of the river. Find (i) the velocity of C in terms of i and j (ii) the velocity of B relative to C in terms of i and j (iii)the magnitude and direction of the velocity of B relative to C (iv)the time it takes B to cross the river (v) PQI, the distance from P to Q. Solution (i) Vc=Oi+3j (ii) Vbc= vb- Vc =5i+9j (iii) I = + 92) = 10.3 Direction = Tan-I (81/25) = E 60.90 N (iv)Time = 100/5 = 20 seconds v) Speed along PQ = •(52 + 122) = 13 m s-l = 260 m 4 c 100 m
  5. 2005 (a) OL Two athletes A and B are running due east in a race. At a certain instant athlete A is x metres from the finishing line and is running with a constant speed of 8 m/s. At this instant athlete B is 6 metres behind A and is running with a constant speed of 10 m/s. B catches up with A at the finishing line, so that the race ends in a dead heat. (i) Find the velocity of B relative to A. (ii) Find the value of x. 2007 OL A river is 72 metres wide and has parallel banks. A boat B departs from point p on the southern bank and lands at point r on the northern bank. The actual velocity Of B is — 4 i+ 3 j m/s. Cyclist C travels due north at a constant speed of 4 m/s across a straight level bridge which spans the river. Find (i) the velocity of C in terms Of i and j. (ii) the velocity of B relative to C in terms of i and j. (iii)the magnitude and direction of the velocity Of B relative to C (iv)the time it takes C to cross the river (v) how much longer it will take B to cross the river. Higher Level 2002 (b) The velocity of ship P relative to a steady wind is 20 km/hr in the direction 800 north of east. c The velocity of ship Q relative to the same steady wind is 10 km/hr in the direction 200 south of west. Calculate the magnitude and direction of the velocity of ship P relative to ship Q. Give your answers to the nearest km and the nearest degree, respectively. 2001 (a) tan Ship B is travellingat 5 34 km/hrin the direction 7 north Of east and ship C is travellingat 5 5 km/hr in the direction tan-1 7 north of west. tan 3 north of east. Show that the speed and direction of ship B relative to ship C is 25 km/hr at 5
  6. 2012 (a) Rain is falling with a speed of 25 m s-l at an angle of 200 to the vertical. A car is travelling along a horizontal road into the rain. The windscreen of the car makes an angle of 320 with the vertical. The car is travelling at 20 m s- Find the angle at which the rain appears to strike the windscreen. 6 25ms
  7. Category 1: Two Independent Bodies (ii) Interception If both bodies have an i and aj component then for interception to occur you must make use of the fact that either the i component or thej component will be the same. 2005 (b) OL A ferry F is travelling due east with a constant speed of 12 km/hr. A boat P is travelling in the direction degrees east Of north with a constant speed of 20 km/hr. At noon P is 1.6 km due south of F and t minutes later P intercepts F. (i) Find the velocity of P relative to F, in terms of i and j and a. (ii) Find the value Of u, correct to the nearest degree. (iii)Find the value of t. Solution F 1.6 km 12 km/hr 20 Vp = 20 sinu i + 20 cosu j [note that a is adjacent to the vertical in this question, where usually it would be adjacent to the horizontal. Can you see why this changes thinks slightly?] (i) 12 i Vpf = (20 sinu i + 20 cosu j) — 12 i Vpf = (20 sinu- 12) i + (20 cos a) j (ii) For interception to occur, the velocity in the i direction must be the same for both (can you explain why - perhaps it would be more helpful if you try to see what would happen if the velocities in the i direction were not the same). i.e. 20 Sina = 12 sinu = 0.6 370 (iii)After t minutes P intercepts F, so in this time P must have moved 1.6 km in the j direction. P 's speed in the j direction (in km/hr) is 20 cosa, so using time = distance/speed t = 1.6/(20 cosa) t = O. I hours (or 6 minutes) 2004 (a) OL Ship A is travelling due north with a constant speed of 15 km/hr. Ship B is travelling north-west with a constant speed of 1542 km/hr. (i) Write down the velocity of ship A and the velocity of ship B, in terms of i and j. (ii) Find the velocity of ship A relative to ship B. (iii)lf ship A is 5.5 km due west of ship B at noon, at what time will ship A intercept ship B? 2008 OL Ship A is 432 km due west of ship B. Ship B is 135 km due west of lighthouse L. A is travelling at a constant speed Of 52 km/h in the direction east ao north, where tan u =5/12. B is travelling due north at a constant speed of 20 km/h. Find (i) the velocity of A in terms of i and j (ii) the velocity of B in terms of i and j (iii)the velocity of A relative to B in terms of i and j. Ship A intercepts ship B after t hours. (iv)Find the value of t. (v) Find the distance from lighthouse L to the meeting point. 7 52 20
  8. 2001 OL 405 knvln• At a certain instant ship D is 16 km due south of ship C. Ship C is travelling with a speed of 442 km/hr in a north-westerly direction. 450 Ship D is travelling with a speed of 41,110 km/hr to intercept C. Let the velocity of D be x i + y j km/hr. (i) Write down the velocity of C in terms of and i and j. (ii) Find the value of x and the value of y. 16 Ian (iii) How long does it take ship D to intercept ship C? knvllr D Higher Level Questions 2010 (a) Two particles, A and B, start initially from points with position vectors 6i - 14j and 3i — 2j respectively. The velocities of A and B are constant and equal to 4i — 3j and 5i — 7 j respectively. (i) Find the velocity of B relative to A. (ii) Show that the particles collide. 1998 (a) The driver of a speedboat travelling in a straight line at 20 m/s wishes to intercept a yacht travelling at 5 m/s in a direction 400 East of North. Initially the speedboat is positioned 5 km South-East of the yacht. Find the direction of the speedboat if it intercepts the yacht and how long the journey takes. 8
  9. Category 1: Two Independent Bodies: (iii) Shortest Distance 2009 OL A ship P is moving north at a constant speed of 20 km/h. Another ship Q is moving south-west at a constant speed of km/h. At a certain instant, P is positioned 50 km due west of Q. Find (i) the velocity of P in terms of i and j (ii) the velocity of Q in terms of i and j (iii)the velocity of P relative to Q in terms of i and j (iv)the shortest distance between P and Q in the subsequent motion. Solution (i) vp=Oi+20j (ii) vq = -1042 cos 45 i - 1042 sin 45 j nvq=-10i-10j (iii) Vpq = + 30 j (iv)Shortest distance 20 50 km Q 10 km/h As we mentioned before, the only way to work out the shortest distance in subsequent motion between two (moving) ships is to make one of the ships seem to be stationary and then analyse the motion of the second ship relative to this first ship. This is what we are doing when we work out Vab. In practice it doesn't matter which ship you make stationary, but sometimes the question is easier to solve one way rather than the other. In this case we have already worked out Vpq (the velocity of P relative to Q) so we are assuming that Q is the stationary ship. So now we draw a diagram and indicate on it the (stationary) ship Q and the initial position of ship P, along with the velocity in which P is travelling relative to Q (i.e. Vpq). In this case we know that Vpq is 10 i + 30 j, so the arrow (originating at P) is going to the right and upwards, and for every 10 units it goes to the right it goes up 30 (so in this case the diagram is not to scale): As we can see from the diagram, the line representing Vpq passes closes to Q when it is perpendicular to a second line which goes P from Vpq to Q (the dotted line in the diagram). 50 VPQ Q This dotted line represents not only the shortest distance between Vpq and Q in the diagram, but also the actual shortest distance between P and Q when both are moving! So now all we need to do is use some simple trigonometry to work out the length of the dotted line. From the diagram we can see that this length corresponds to 50 sinu, so first we need to work out Cl. Remember that in order to work out the direction of Vpq we used the fact that Vpq is 10 i + 30 j and for every 10 units it goes to the right it goes up 30. so a = tan-1(30/10) = 71.570 Shortest distance = 50 sin = 50 sin (71.570) = 47.43 km 9 30 10
  10. 2003 OL The velocity of ship A is 3 i — 4 j mis and the velocity Of ship B is — 2 i + 8 j m/s. (i) Find the velocity of ship A relative to ship B in terms of i and j. (ii) Find the magnitude and direction of the velocity of ship A relative to ship B, giving the direction to the nearest degree. At a certain instant, ship B is 26 km due east of ship A. (iii)Show, on a diagram, the positions of ship A and ship B at this instant and show, also, the direction in which ship A is travelling relative to ship B. (iv)Calculate the shortest distance between the ships, to the nearest km. 2006 OL Ship A is travelling east (10 north with a constant speed of 39 km/h, where tan = 5/12. Ship B is travelling due east with a constant speed of 16 km/h. At 2 pm ship B is positioned 90 km due north of ship A. (i) Express the velocity of ship A and the velocity of ship B in terms of i and j. (ii) Find the velocity of ship A relative to ship B in terms of i and j. (iii)Find the shortest distance between the ships. 10 B 16 knVh 90 km 39 knvh
  11. Shortest Distance - Higher Level Questions Usually at higher level one of the cars will have to be brought to the junction — so which one? Technically it doesn 't matter, but the solution is more straightforward if (when one car is at the junction) the other is also on the x—axis. 2009 (a) Two cars, A and B, travel along two straight roads which intersect at right angles. A is travelling east at 15 m/s. B is travelling north at 20 m/s. At a certain instant both cars are 800 m from the intersection and approaching the intersection. (i) Find the shortest distance between the cars Solution (i) 15i+Oj Vb=Oi+20j Vab= 15i-20j Magnitude = •(152 + 202) = 25 m/s {we will need this later} = tan-I (20/15) = 53.130 Now we bring B to the junction by 'running the video forwards'. B reaches the intersection in 800/20 = 40 seconds. In this time A travels 15 x 40 = 600 m and is 'now' 200 m from the intersection. So now we re-draw our position diagram and on it indicate the velocity of A relative to B: 200 53.13 x 25 From the diagram we can see that the shortest distance (the dotted line) corresponds to 200 Sina = 200 sin (53.130) = 160 m 11
  12. 2011 (a) Two cars, A and B, travel along two straight roads which intersect at an angle 9 where tan 9 = ± . Car A is moving towards the intersection at a uniform speed of 5 m s-l. Car B is moving towards the intersection at a uniform speed of 10 m s-l. At a certain instant each car is 100 m from the intersection and approaching the intersection. Find (i) the velocity of A relative to B (ii) the shortest distance between the cars. 1979 A ship A is travelling South-West at 10 knots. Another ship B is travelling at 20 knots in a direction 300 North of West. Draw a diagram to show the velocity of B relative to A. Calculate the magnitude of the relative velocity, correct to the nearest knot, and its direction correct to the nearest degree. By how much should A increase its speed, without changing direction, so that B would appear to A to be travelling due North? 1982 Two straight roads intersect at an angle of 60 Cyclists A and B move towards the point of intersection at 30 km/h and 40 km/h, respectively. (i) Calculate the velocity of A relative to B. (ii) If A is 3.5 km and B is 2 km from the intersection at a given moment, calculate the shortest distance between them in their subsequent motion. 1986 Two straight roads intersect at an angle 0, tan 0 = 4/3. Cars A and B move towards the point of intersection at 16 m/s and v m/s, respectively. If the magnitude of the velocity of A relative to B is 16 m/s, find v. If at a given instant A is 96 m and B is 38.4 m from the intersection, calculate (i) the shortest distance between them in their subsequent motion (ii) the distance, to the nearest metre, between the two cars 2 seconds before the instant when are nearest to each other 2006 (b) A boy swims due west at a speed of 0.8 m/s A girl swims at 0.4 m/s in the direction ao south of west. At a certain instant the girl is 10 m, 600 north of west of the boy and 10 s later she is due north of the boy (i) Find the distance travelled by the boy and the girl in 10 s. (ii) Hence, or otherwise, find the value of a. (iii)Find the shortest distance between the boy and the girl in the subsequent motion. 12 600 10m 0.4 m/s 600 0.8 m/s
  13. 2004 (b) At time t = O, two particles P and Q are set in motion. At time t = O, Q has position vector 20 i + 40 j metres relative to P. P has a constant velocity of 3 i + 5 j m/s and Q has a constant velocity Of 4 i — 3 j m/s. (i) Find the velocity of Q relative to P. (ii) Find the shortest distance between P and Q, to the nearest metre. (iii)Find the time when P and Q are closest together, correct to one decimal place. 2000 At a certain instant ship Q is at a distance of 4a due east of ship P. Q is moving northwards with constant speed u and P is travelling with constant speed 2u. Find the direction of P if it is to intercept Q. Find the time T, in terms of a and u, it would take P to intercept Q. If, instead, after time — has elapsed, the speed of P drops to constant speed u, without changing direction, find, in terms of a, (i) the shortest distance between P and Q (ii) the distance each ship has moved from its original position to its position when they are closest together. 1999 (b) Two ships A and B move with constant speeds 48 km/h and 60 km/h respectively. At a certain instant ship B is 30 km west of A and is travelling due south. Find (i) the direction A should steer in order to get as close as possible to ship B (ii) the shortest distance between the ships. 13
  14. Category 1: Two Independent Bodies — (iv) Distance from Intersection (Higher Level) 2008 (a) Two straight roads cross at right angles. A woman C, is walking towards the intersection with a uniform speed of 1.5 m/s. Another woman D is moving towards the intersection with a uniform speed of 2 m/s. C is 100 m away from the intersection as D passes the intersection. Find (i) the velocity of C relative D (ii) the distance of C from the intersection when they are nearest together. 2009 (a) Two cars, A and B, travel along two straight roads which intersect at right angles. A is travelling east at 15 m/s. B is travelling north at 20 m/s. At a certain instant both cars are 800 m from the intersection and approaching the intersection. (i) Find the shortest distance between the cars (see previous section for a worked solution to this part). (ii) Find the distance each car is from the intersection when they are nearest to each other. 2005 (b) Two straight roads intersect at an angle of 450. Car A is moving towards the intersection with a uniform speed of p m/s. Car B is moving towards the intersection with a uniform speed of 8 m/s. The velocity of car A relative to car B is —2i — 10 j, where i and j are unit perpendicular vectors in the east and north directions, respectively. At a certain instant car A is 22002 m from the intersection and car B is 136 m from the intersection. (i) Find the value of p. (ii) How far is car A from the intersection at the instant when the cars are nearest to each other? Give your answer correct to the nearest metre. 1990 Two straight roads intersect at an angle 600. Car A moves towards the junction with uniform speed 16 m/s, while car B moves away from the junction with uniform speed 20 m/s. Calculate the velocity of A relative to B If A is 450 m and B is 200 m from the intersection at a given moment, calculate the time 450 interval in seconds until the cars (i) are nearest to each other (ii) are equidistant from the intersection. 2003 (b) Two straight roads intersect at an angle of 600. Car A is moving towards the intersection with a uniform speed of 7.5 m/s. Car B is moving towards the intersection with a uniform speed of 10 m/s. Car A is 375 m away from the intersection as car B passes the intersection. (i) Find the velocity of car A relative to car B. (ii) How far is each car from the intersection at the instant when they are nearest to each other? Give your answers correct to the nearest metre. 14 600
  15. 2004 (a) A cyclist A is pedalling at 3 m/s due east along a straight road. A second cyclist B is pedalling at 4 m/s due north along another straight road intersecting the first at a junction p. If A is 80 m and B is 40 m from p at a given moment, calculate (i) the velocity of B relative to A. (ii) how far each cyclist is from p when they are nearest together. 2004 (b) If when A and B are 80 m and 40 m from p, respectively, then A immediately accelerates at 0.1 m/s2 and B decelerates at q m/s2. (i) Find the velocity of B relative to A in terms of time t. (ii) Determine the value of q which causes them to arrive at p together. 1995 (a) Two particles A and B are moving along two perpendicular lines towards a point O with constant velocities of 6 m/s and 8 m/s respectively. When A is 64 m from O, B is 62 m from O. (i) Find the distance of each particle from O after t seconds. (ii) Hence, or otherwise, find the times at which their distance apart is 50 m. 15 8
  16. Two Independent Bodies — (v) Distance from each other To solve this one draw a circle around the 'stationary' Object whose radius corresponds to the distance given in the question (say 10 km). Now draw the V ab vector. Where this cuts the circle corresponds to A first coming within 10 km of B and where it cuts the circle on the way out corresponds to A being more than 10 km of B. Use Pythagoras' Theorem to get the length Of this chord. Now knowing this length and the velocity Vab, work out the time. 2012 (b) At noon ship A is 50 km north of ship B. Ship A is travelling southwest at 24 02 km h- Ship B is travelling due west at 17 km h- (i) Find the magnitude and direction Of the velocity of B relative to A. A and B can exchange signals when they are not more than 20 km apart. (i) At what time can they begin to exchange signals? How long can they continue to exchange signals? 2007 (a) Ship B is travelling west at 24 km/h. Ship A is travelling north at 32 km/h. At a certain instant ship B is 8 km north-east of ship A. (i) Find the velocity of ship A relative to ship B. (ii) Calculate the length of time, to the nearest minute, for which the ships are less than or equal to 8 km apart. 2002 (a) Two boats, B and C, are each moving with constant velocity. At a certain instant, boat B is 10 km due west of boat C. The speed and direction of boat B relative to boat C is 2.5 m/s in the direction 600 south of east. (i) Calculate the shortest distance between the boats, to the nearest metre. (ii) Calculate the length of time, to the nearest second, for which the boats are less than or equal to 9 km apart. 1987 At a certain instant a ship H is 37.5 km due West of a ship K. Ship H is travelling South-East at 25 km/h and ship K is travelling South at 15 km/h. (i) Draw a diagram to show the velocity of K relative to H and calculate the magnitude and direction of the relative velocity. (ii) If H and K can exchange signals when they are not more than 20 km apart, calculate when they can begin to exchange signals and for how long they can continue to exchange signals. 1984 A ship B is travelling in a direction 410 East of North at 15 m/s. A second ship C is travelling 410 South of East at 20 m/s. Calculate (i) the velocity of B relative to C (ii) the shortest distance between the ships if C is 3 km East of B at a particular moment (iii)the time interval during which the ships remain in visual contact, if visibility is limited to 3 km. 16
  17. 1985 (a) Two cars A and B are moving along straight roads which are at right angles to each other, with uniform velocities 3 m/s and 4 m/s, respectively. When B is at the crossroads, A is away. Calculate the time interval for which the distance between the cars is not greater than 82 m. 1996 {Tricky) A ship, B is travelling due West at 25.6 km/h. A second ship, C, travelling at 32 km/h is first sighted 17 km due North of B. From B the ship C appears to be moving South-east. Find (i) the direction in which C is actually moving (ii) the velocity of C relative to B (iii)the shortest distance between the ships in the subsequent motion (iv)the time that elapses, after first sighting, before the ships are again 17 km apart. 17
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  19. Category 2: Body and Carrier For these questions information will be given or be required for each of the following: l. the velocity of the body (Vb) 2. the velocity of the carrier (Vc) 3. the velocity of the body relative to the carrier (Vbc) The velocity of the body relative to the carrier is usually either the velocity of a plane relative to the wind or velocity of a boat relative to the river. There are various ways of announcing the velocity of a body relative to the carrier (Vbc) l. "The velocity Of the body relative to the carrier" (Duh!) 2. ' 'The velocity Of the body in still air" 3. ' 'The velocity at which it starts" (i.e. before the wind or river kicks in) Each Of points two and three would take quite a while to explain using diagrams so if you can 't figure it Out for yourself you'll just have to take it for granted (which is never ideal I know). There are various ways of announcing V b 2. 3. 4. 'The velocity Of the body" ' 'The final or actual velocity Of the body" "The velocity Of the body as observed from the ground" "The velocity Of the body after taking the carrier into account" Key to answering questions on this section It is (almost) always easier to use a diagram rather than vector analysis. It can be tricky to get the hang of these diagrams but once you do the questions are almost always very straightforward and just as importantly can be answered very quickly. Points to note: Sometimes it is difficult to establish an angle in the velocity diagram. In these cases the angle can almost always be obtained by drawing a separate position diagram. If drawing both on the same diagram do not get them confused - Each diagram is either a velocity diagram or a position diagram — one diagram cannot be both, i.e. don't express an angle as being the sin of a distance over a velocity (see 1995 (a) and 1998). On other occasions when finding it hard to establish an angle it is very useful to extend the lines (lightly) on all sides beyond the vertices. This often makes it easier to see connections. 19
  20. Drawing the diagram The diagram will consist of a triangle where each of the sides represents one of the following: l. the velocity of the body (Vb) 2. the velocity of the carrier (Vc) 3. the velocity of the body relative to the carrier (Vbc) Golden rule: you need two sides and an included angle Procedure: . You will be told the direction of either Vb or Vbc — begin by drawing in this vector. 2. The carrier vector. draw this vector on its own off to the side initially. 3. Note that the question may well tell you the direction that this vector is coming from — you need to use the direction that it is going towards. 4. It is often useful to draw an x-y axis on both ends of this vector to establish a relevant angle for the main diagram. 5. Now knowing the direction of the carrier should enable you to establish the direction of the unknown quantity ( Vb or Vbc) — put this in now. 6. Note that the vectors Vb and Vbc both originate from the same point. Finally use the Vc vector to complete the triangle (it goes from the head of Vbc to the head Of Vc). 7. If the body "crosses as quickly as possible" it means that the body begins by going straight across (which in our diagram would be Vbc). 20
  21. Body and Carrier exam questions River and carrier 1993 (b) A boy who can swim at 5/9 m/s wishes to cross a river 50 m wide, flowing at 5/6 m/s, as quickly as possible. Calculate (i) the direction he should take. (ii) the time he takes to cross. (iii)how far downstream from his starting point he goes. 1995 (b) A girl wishes to swim across a river 60 m wide. The river flows with a velocity of q m/s parallel to the straight banks and the girl swims at a velocity of p m/s relative to the water. In crossing the river as quickly as possible she takes 100 s and is carried downstream 45 m. Find (i) p and q. (ii) how long will it take her to swim in a straight line back to the original starting point. 1988 (a) Two boats move with constant speed 5 m/s relative to the water and both cross a straight river of width 72 m flowing with constant speed 3 m/s parallel to the banks. One crosses by the shortest path and the other in the shortest time. Show that the difference in the times taken is 3.6 s. 2000 (b) OL A river is 100 m wide and is flowing with a speed of 2 m/s parallel to the straight banks. The speed of a swimmer in still water is 3 m/s. (i) What is the shortest time it takes the swimmer to swim across the river? (ii) What direction should the swimmer take so as to swim straight across to a point directly opposite? How long will it then take the swimmer to cross to this point? 1998 (b) A man wishes to row a boat across a river to reach a point on the opposite bank that is 25 m downstream from his starting point. The man can row the boat 3.2 m/s in still water. The river is 45 m wide and flows uniformly at 3.6 m/s Find (i) the two possible directions in which the man could steer the boat (ii) the respective crossing times 2011 (b) A woman can row a boat at 4 m s-l in still water. She rows across a river 80 m wide. The river flows at a constant speed of 3.5 m s-l parallel to the straight banks. She wishes to land between B and C. The point B is directly across from the starting point A and the point C is m downstream from B. If 9 is the direction she takes, find the range of values of 9 if she lands between B and C. 21 c 80 m 4
  22. 2007 (b) A man can swim at 3 m/s in still water. He swims across a river of width 30 metres. He sets out at an angle of 300 to the bank. The river flows with a constant speed of 5 m/s parallel to the straight banks. In crossing the river he is carried downstream a distance d metres. Find the value of d correct to two places of decimals. 2005 (a) 30 5 3 30 A woman can swim at u m/s in still water. She swims across a river of width d metres. The river flows with a constant speed of v m/s parallel to the straight banks, where v < u. Crossing the river in the shortest time takes the woman 10 seconds. Find, in terms of u and v, the time it takes the woman to cross the river by the shortest path. 1980 A boat has to travel by the shortest route to the point 4.25 j km and then return immediately to its starting point at the origin. The velocity of the water is (8•.15 — —j ) km/hour and the boat has a speed of 18 km/hour in still water. (i) If a i + b j is the velocity of the boat on the outward journey, find a and b and the time taken for the outward journey, leaving your answer in surd form. (ii) Find, also, the time taken for the whole journey. 1975 A man wishes to swim across a river 60 m wide. The river flows with a velocity of 5 m/s parallel to the straight banks and the man swims at a speed of 3 m/s relative to the water. If he heads at an angle to the upstream direction, and his actual velocity is at an angle O to the downstream 3sina direction, show that tan O = 5—3cosa . Prove that tan 0 has a maximum value when cos = 3/5. Deduce that the time taken for the man to cross by the shortest path is 25 s. 22
  23. Wind and carrier 2004 (a) A bird flies at a uniform speed of 22 m/s. It wishes to fly to its nest which is 250 m due north of its present position. There is a wind blowing from the southeast at 18 m/s. (i) Find the direction, to the nearest degree, in which the bird must fly to reach its nest (ii) Find the time taken to reach the nest, correct to two decimal places. 1999 (a) An aeroplane has a speed of 160 m/s in still air. When the wind blows from the east, the velocity of the aeroplane as observed from the ground is 120 m/s towards the north-east. Find the speed of the wind correct to two decimal places. 1992 An aeroplane having a speed of 500 km/h in still air travels 1500 km due North when the wind is blowing from 600 East of North at 90 km/h, and then returns to the starting point along the same path. Calculate (i) the directions in which the aeroplane must travel on the outward and return journeys. (ii) the total time taken (iii)the total time taken if there was no wind blowing. 2001 (b) The speed of an aeroplane in still air is 160 km/hr. It flies in a straight line from p to q and back again. Point q is due north of point p. Throughout the journey there is a wind blowing from the south-west at 32 km/hr. The time for the whole journey is 5 hours. Find the distance from p to q. Give your answer to the nearest km. 2006 (a) Two aeroplanes A and B, moving horizontally, are travelling at 200 km/h relative to the ground. There is a wind blowing from the east at 60 km/h. The actual directions of flight of A and B are north-west and north- east respectively. (i) Find the speed of aeroplane A in still air. (ii) Find the magnitude and direction of the velocity of A relative to B. 1983 An aircraft flew due east from p to q at ul km/h. Wind speed from the south west was v km/h. On the return journey from q to p, due west, the aircraft's speed was km/h, the wind speed being unchanged. If the speed of the aircraft in still air was x km/h, x v, show, by resolving along and perpendicular to pq, or otherwise, that (i) ul —u2=v (iii)lf V•ql = d, find in terms of v, x and d, the time for the two journeys. 2009 (b) {very difficult} The speed of an aeroplane in still air is u km/h. The aeroplane flies a straight-line course from P to Q, where Q is north of P. If there is no wind blowing the time for the journey from P to Q is T hours. Find, in terms of u and T, the time to fly from P to Q if there is a wind blowing from the south-east with a speed of 4 €2 km/h. 23
  24. Category 3: 'The wind appears to blow' (Wind Appears) Introduction If you are told that"the wind appears to be " this represents Vwo (the velocity of the wind relative to the observer) not Vow. Answering questions Here we use vector analysis. An observer walks in one direction (at a given speed) and notices the wind to have a certain velocity. He then changes direction (and speed) and as a result notices that the wind seems ('appears') to have a different velocity. The question is 'What is the actual velocity Of the wind?' To answer this we make use of the fact that the true velocity of the wind (V w) is the same in both scenarios, so we get an expression for Vw (in terms of the velocity of the observer (Vo) and the apparent velocity Of the wind (Vwo)) for both scenarios and let one expression equal the other. i.e. let Vw = Vwp + Vo, where V w will have the same value in both scenarios. Watch out for the following (tricky) phrases 'The wind blows towards the East (this one is obviously straightforward) Key: vw = Xi "The wind comes from the East; (i.e. the wind is going towards the West) Key: vw = -Xi "The wind comes from the South-East"; (i.e. the wind is going towards North-West) Key: vw = -Xi + xj "The wind blows from the direction 2i + 3j; Key: vw = -2Xi - 3Xj (seei991) Note the variation to this question in 2008. See the 'guide to answering questions' for a suggested approach. 24
  25. Exam Questions 1977 To a cyclist riding North at 7 m/s the wind appears to blow from the North-West. To a pedestrian walking due West at I m/s the same wind appears to come from South-West. Find the magnitude and direction of the velocity of the wind by expressing it in the form direction u i + v j, or othenvise. 2003 (a) A woman travelling north at 10 km/h finds that the wind appears to blow from the West. When the woman trebles her speed, the wind appears to blow from the North-West. Find the velocity of the wind. 1993(a) A girl travelling south at I I m/s finds that the wind appears to blow from the East but on doubling her speed it appears to come from the South-East. Calculate the magnitude and direction Of the wind's velocity. 2010 (b) When a motor-cyclist travels along a straight road from South to North at a constant speed Of 12.5 m s-l the wind appears to her to come from a direction North 450 East. When she returns along the same road at the same constant speed, the wind appears to come from a direction South 450 East. Find the magnitude and direction of the velocity of the wind. 1989 A man travelling North at 20 m/s finds that the wind appears to blow from the West. When he travels due West at 8.45 m/s the wind appears to blow from the South West. (i) Calculate the velocity of the wind. (ii) If the man travelled in a direction 300 North of West at 8 m/s from what direction would the wind appear to blow? 1991 A runner observes that when her velocity is u i the wind appears to come from the direction 2 i + 3 j , but when her velocity is u j the wind appears to come from the direction - 2 i + 3 j . 511 (i) Prove that the true velocity of the wind is 6 i - 4 J' (ii) Find the speed and direction of motion of the runner when the wind velocity appears to be u i . 2008 (b) On a particular day the velocity of the wind, in terms of i and j, is x i — 3 j where x E N. i and j are unit vectors in the directions East and North respectively. TO a man travelling due East the wind appears to come from a direction North West where tan a = 2. When he travels due North at the same speed as before, the wind appears to come from a direction North 130 West where tan = 3/2. Find the actual direction of the wind. 25
  26. 1997 To a man walking North at 4 m/s a constant wind appears to have a speed of 3 m/s. The man then changes direction and when he is walking West at 3 m/s the wind appears to have a speed of 4 There are two possible wind velocities. If i and j are unit vectors in the East and North directions, respectively (i) show that — 3 i + 4 j is one possible wind velocity, (ii) find the second possible wind velocity. 26
  27. 2010 (i) vc=Oi+3j (ii) Vbc=5i+9j (iii) I Vbcl = 10.3, direction = (iv)Time = 20 s (v) Speed = 13 m 2009 (i) Vp=Oi+20j (ii) vv=- 10i- 10j (iii)Vpo = + 30 j (iv)Shortest Distance — 2008 (i) VA =48i+20j (ii) VB = Oi +20 j (iii)VAB = 48 i (iv)Time = 9 hours Answers to Ordinary Level Exam Questions E 60.90 N _ 47.43 km (v) Distance = 225 km 2007 (ii) VBC = -4 i- Ij (iii)Speed = 4.12 m s-l, O = 14.040 with the bank (iv)time = 18 s (v) time = 24 s required time = 6 s 2006 (i) VA = 36 i + 15 j VB = 16 i + 10 j (ii) VAB =20i+ 15 j (iii)Shortest Distance = 72 km 2005 (a) (i) VBA =2i (ii) x = 24 m 2005 (b) (i) Vpf= (20 Sina - 12) i + 20 cos a j (ii) = 370 (iii)t = 0.1 h or 6 minutes 2004 (a) (i) VA =Oi+ 15 j VB=-15i+15j (ii) VAB = 15 i +0 j (iii)Time = 12:22 2004 (b) (ii) Distance = 65 m 27
  28. 2003 (i) VAB=5i- 12 j (ii) Magnitude = 13 m s-l Direction = 670 south of east (iii) (iv)Shortest distance = 24 km 2002 (i) VA = -IOi+Oj (ii) (iii)VAB = 5 i + 12 j (iv)Speed = 13 km/hr, Direction = 670 N of w 2001 (ii) y = 12 (iii)t = 2 2000 (a) VAB = 15i +20j 2000 (b) (i) time = 100/3 s (ii) Direction: sin = 2/3 (iii)time = 100/45 s 28
  29. Answering Higher Level Exam Questions Calculate the length of time for which the objects are less than or equal to 10 km apart. Calculate the time it takes to get to the shortest distance point A common mistake here is to divide the shortest distance by VAB instead of dividing the relative distance travelled by VAB. Can you see why this would be wrong? 2009 (a) (i) Straightforward. Bring B to junction first; A is now 200 m from junction. Then use normal procedure to get shortest distance = 160 m. (ii) Straightforward. Time to get to shortest distance = 0.8 + 40 seconds. Use this to work out distance of each from intersection. Ans: A = 128 m, B = 96 m past the intersection. 2009 (b) Nasty, nasty question. Begin as normal by drawing a diagram and use the sin rule to get u sin = 4 (where is the angle between s and u). Use the expression time = distance/velocity, but because the distance corresponds to the distance from P to Q (vertical distance) the velocity must be in the same direction. Now comes the tricky bit; we don't have a vertical velocity so we must add the vertical component Of u to the vertical component of 442 to calculate the velocity in the direction P—+Q (hopefully this will be easier to understand when referring to the diagram (see marking scheme). 2008 (a) Straight-forward "Two-Independent-B0dies" type question. (i) Ans: VCD= 1.5i—2j. (ii) See 2003 (b). Draw a relative velocity diagram keeping D at the junction. To find out how long it takes C to get to the shortest distance position divide 100 Cos O by VCD to get 24 secs. Once you have the time from this relative velocity diagram, you then apply this to the real-world situation to ascertain how far C has travelled and from this how far C would be from the intersection at this stage. Ans: Distance of C from intersection = 64 m. 2008 (b) This is a "Wind Appears to Blow" type question, but with a twist. Usually for "wind appears" type questions you are told the velocity Of the man and the velocity Of the wind relative to the man, and from this you get an expression for the velocity of the wind. Then repeat this for the second scenario. In this question you are told the velocity of the wind at the beginning (v w = xi — 3j), along with the direction in which the man is travelling (Vm = ai). Now put these together to get an expression for the velocity Of the wind relative to the man (Vwm). In this case we get Vwm = (x-a)i — 3j. Now because you are told that the wind appears to come from a direction North a 0 West where tan a = 2, you use this to say Tan = (Vj/Vi), i.e. 2 = Vj/Vl, i.e. 2 = 3/(x-a). Repeat for the second scenario to get two equations and solve. Note that there is an extra twist to this question: We usually say "Tan = (Vj/Vi)", but this only applies where the angle is given with respect to the x-axis. In this case the angle was given with respect to the y-axis (North (10 West means first go North, then go towards the left). so instead Of saying "Tan = (VJ/Vi)", we need to say that 'Tan = (Vi/Vj)". Can you see why? This gives us for the first scenario above: 2 = (x- a)/3. Use the same approach for the second scenario. Ans: direction of the wind is North 71.60 West 29
  30. 2007 (a) Straight-forward ' 'Two-Independent-B0dies" type question. (i) Easy peasy. Ans: VAB = 24 i + 32 j. (ii) "Calculate the length of time for which the boats are less than or equal to 8 km apart" (see also 2002 (a)). Draw a diagram, then draw a circle of radius 8, with the centre at A, and use geometry to solve. Ans: time = .39 hours or 24 minutes. 2007 (b) Body-and-carrier type question. First draw an accurate diagram putting in all relevant information. Next find how long it takes to cross (30 / 3 Sin 30) and put this off to the side. Now from the diagram the net velocity will be 5 — 3 Cos 30. Distance is then = vel x time. Ans: Distance = 48.04 m. Note that you must give it to 2 decimal places. 2006 (a) (i) Fairly straight-forward body-and-carrier type question. Make sure to draw a diagram to help you. Remember when it asks for speed it means magnitude. Ans: v = 163.2 km/hr (ii) Ans: Magnitude = 282.4 km/hr, direction is due west. 2006 (b) This was a very difficult question. Just drawing the diagram representing all the available information was in itself 'non-trivial' (as maths people like to say). (i) Easy peasy; boy travelled 8 m, girl travelled 4 m. (ii) Use geometry to show that 8 = 10 Cos 60 + 4 cos (1 Ans: a = 41.410. (iii) Draw in the VGB line and use geometry again to find any necessary angles. Ans: Shortest distance = 5.3 m. 2005 (a) Fairly straight-forward body-and-carrier type question You are told that crossing the river in the shortest time takes 10 seconds. Use this to get d = 10 u. Now draw an appropriate diagram and label it, then use Pythagoras to get the velocity straight across as 2005 (b) "Two-Independent-B0dies" type question. Straight-forward. Ans: p = 1002. (ii) Find out how long it takes A to reach the intersection then find out how far B has travelled in this time and from that how far B is from the intersection. Now draw a relative-velocity diagram and use the adjacent to find how long it took A to get to the shortest distance point. Now that you know the time you can apply this to the real-world situation and combine it the real-world velocity to get the actual distance of A from the intersection. Ans: Il m. 2004 (a) Fairly straight-forward body-and-carrier type question. Ans: direction = 550 North of East. (ii) Straightforward. Time = 8.16 secs 30
  31. 2004 (b) Straightforward ' 'Two-Independent-B0dies" type question. (i) Vqp = i — 8 j (ii) Shortest distance = 25 m. (iii) Time = 4.6 secs. 2003 (a) Very straightforward "Wind Appears to Blow" type question. Ans: w = 20 i + 10j 2003 (b) Straightforward. VAB = 2.5 i - j. (ii) Draw a relative velocity diagram keeping B at the Junction. To find out how long it takes A to get to the shortest distance position divide 375 Cos O by VAB. Once you have the time from this relative velocity diagram, you then apply this to the real-world situation to ascertain how far each has travelled and from this how far each would have travelled from the intersection at this stage. Ans: Distance of A from intersection = 288 m, distance of B from intersection = 115 m. 2002 (a) Straightforward. Ans: Shortest distant = 8,660 m (ii) "Calculate the length of time for which the boats are less than or equal to 9 km apart" (see also 2007 (a)). Draw a diagram, then draw a circle of radius 8, with the centre at A, and use geometry to solve. Ans: Time = 8,660 secs. 2002 (b) This one was particularly nasty. You were given Vpw and v qw and asked to find V Here you must use the fact that Vpw = Vp — Vw, SO Vp = V PW + Vw Vqw = Vq — Vw, SO Vq = V qw + Vw Now because Vpq = Vp — Vq, when you use the expressions above for Vp and Vq, the Vw cancels out in each and you are left with Just the Vpw and the Vqw to work with. Don't forget about the magnitude and direction. Ans: Magnitude = 26 km/hr and direction is 610 North of East. 2001 (a) Straightforward 2001 (b) Fairly straightforward body-and-carrier type question. The last part is a bit tricky; if total time taken is two hours, it means d/V1 + d/V2 = 5. Ans: d = 388 km. 31
  32. N/A
  33. 2000 The first section (two questions) was okay; the second section was very testing. Direction of P: For interception to occur the j components must be equal, i.e. 2 u Sina = u, O = 30 . To calculate time use velocity x time = 4a, i.e. 2u cosu (T) = 4a, T = 4a/uN13 Now here is where it starts to get tough. First off, go back to the beginning and this time draw in the motion of both ships but stop after time T/2 has elapsed. P and Q are now (still) on the same horizontal line and Q is a distance 2a west of P But from now on the speed of P = u, i.e. Vp = u cos 30 i + u sin 30 j and Vq = 2u j. Proceed as normal to get shortest distance. Answer = a (ii) Proceed as normal here also. First calculate the time to reach the shortest distance point (t = aQ3/2). Next Multiply this time by the respective velocities to calculate the distance travelled by each ship for the second stage and remember to add on the distance travelled during the first stage to each. Answer: distance travelled by P = 7a/Q3 and distance travelled by Q = 50/03. Phew. 1999 (a) Straightforward if you draw a diagram. Answer: v = 50.79 m/s. 1999 (b) I considered this question to be ridiculously difficult, but that' s just because I couldn't do it and maybe I over-estimate my own ability at times. Feel free to crack it and let me know how. (i) Answer: (1 = sin-1 4/5 (ii) Shortest distance = 18 km. 1998 (a) (i) Tricky. Draw a velocity diagram but use position of the boats to give information about angles in the velocity diagram. Remember when using the sin rule that you can't mix velocity and displacements. Note also that the displacement is in km — nasty. Answer: direction is W 59.420 N (ii) Straightforward. Answer: Time = 264.13 s 1998 (b) (i) This is rather counter-intuitive in that you would imagine that the man could only steer the boat in one direction to get to the desired point on the far bank. However proceed as normal and when you use the sin rule note that due to symmetry about 900 if = 79.550 it could also equal 100.450. When setting up the diagram as per part (a)use information from the position diagram to give angles in the velocity diagram. Answer: = 39.50 and 18.60. (ii) Straightforward. Answer: time 1 = 22.1 seconds and time 2 = 1997 44.1 seconds. 33
  34. It's a full question and a fairly testing one. Normally 'wind-appears' type questions are ones which you would rely on to pick up almost full marks, but it's not a gimme here. You're given Vm, let Vw = xi + yj and then express Vwm in terms of these two. Now because you are told that the 'speed' is 3 m/s you must express Vwm as a magnitude using Pythagoras. Square both sides to get an equation with x and y. Repeat for the second situation and solve (note that there is still quite a bit of algebra still to do at this stage). Answer: V w = 0.84 i + 1.12 j 1996 Full question (i) Let vcb=Xi-xj. We also know + 25.6i. Equate and get Vc on its own vith everything else on the other side. Now let Vc = 32 and let this equal the magnitude of the Other side. We now have an equation with just x 's in it so can solve and then get a direction for Vc. Answer: Direction = 79.450 South of East. (ii) Straightforward. Answer: Vcb = 31.46 i — 31.46 j. (iii)Straightfonvard once you draw a diagram. Answer: shortest distance = 12.02 km. (i v) Straightforward. Answer: time = 0.54 hours. 34

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