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Question 2: Relative Velocity Please remember to photocopy 4 pages onto one sheet by going A3→A4 and using back to back on the photocopier
Page
Overview 2
Category 1: Two Independent Bodies.
Introduction 2 (i) General Questions 4
(ii) Interception 6
(iii) Shortest Distance 8
(iv) Distance from Intersection 13
(v) Distance from each other (^) 15
Category 2: Body and Carrier
Introduction 17
Exam questions 19
Category 3: Wind Appears Introduction 22
Exam Questions 23
Answers to Ordinary Level Exam Questions 25
Guide to answering Higher Level Exam Questions 2009 - (^1996) 27
************* Higher Level Marking Schemes to be provided separately ***************
Overview Relative Velocity questions can be sub-divided into three categories:
- Two Independent Bodies
- Body and Carrier
- 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-1. Another car (car A) is following behind you travelling at 25 m s-1. 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-1. We can say that the velocity of car B relative to car A is 5 m s-1. We write this mathematically as follows: Vab = 5 m s-1. 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-1^ and car B was following at a velocity of 20 m s-1^ then Vab = - 5 m s-1, i.e. the velocity of A relative to B is minus 5 m s-1, which means that as car B (you) see it, car A is moving backwards at a speed of 5 m s-1.
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-1, i.e. the occupants of car A see car B moving ahead of them at a speed of 5 m s-1.
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- while car A is following behind you travelling at 25 m s-1. As before, Vab = 5 m s-1. 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.
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 Vab = Va – Vb = (15 i) – (- 20 j) = 15 i + 20 j
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) = 10 i + 0 j
(ii) Distance = 100 + Sq – Sp = 100 + 10(3.5) – 20(3.5) = 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-1. Cyclist C travels due north at a constant speed of 3 m s-1^ 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) | PQ |, the distance from P to Q. Solution (i) Vc = 0 i + 3 j
(ii) Vbc = Vb – Vc = (5 i + 12 j) – (0 i + 3 j) = 5 i + 9 j
(iii)Vbc = √(5^2 + 9^2 ) = 10. Direction = Tan-1^ (81/25) = E 60.9^0 N
(iv) Time = 100/5 = 20 seconds
(v) Speed along PQ = √(5^2 + 12^2 ) = 13 m s- PQ= (20)(13) = 260 m
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.
2002 OL
Ship A is travelling due west with a constant speed of 10 km/hr. Ship B is travelling at a constant velocity. At 1200 hours, the radar screen of ship A shows the position of ship B relative to ship A as -2 i -20 j km. At 1400 hours, two hours later, the position of ship B relative to ship A is 8 i + 4 j kilometres. (i) Write down the velocity of ship A in terms of i and j. (ii) Show that the change in the position of ship B relative to ship A between 1200 hours and 1400 hours is 10 i + 24 j. (iii)Find the velocity of ship B relative to ship A. (iv) Find the speed and direction of ship B. Give the direction to the nearest degree.
Higher Level 2002 (b) The velocity of ship P relative to a steady wind is 20 km/hr in the direction 80° north of east. The velocity of ship Q relative to the same steady wind is 10 km/hr in the direction 20° 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)
Ship B is travelling at 5 34 km/hr in the direction
tan 1 6 7
− north of east and ship C is travelling at 5 5
km/hr in the direction tan −^17 north of west.
Show that the speed and direction of ship B relative to ship C is 25 km/hr at
tan 1 1 3
− north of east.
Category 1: Two Independent Bodies (ii) Interception If both bodies have an i and a j component then for interception to occur you must make use of the fact that either the i component or the j 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 α 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 α. (ii) Find the value of α, correct to the nearest degree. (iii)Find the value of t. Solution Vp = 20 sinα i + 20 cosα j [note that α 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) Vf = 12 i Vpf = (20 sinα i + 20 cosα j) – 12 j Vpf = 20 sinα i + (20 cos α - 12) 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 sinα = 12 sinα = 0. α = 37^0
(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 cosα, so using time = distance/speed t = 1.6/(20 cosα) t = 0.1 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 15√2 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)If 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 α^0 north, where tan α =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.
2001 OL
At a certain instant ship D is 16 km due south of ship C. Ship C is travelling with a speed of 4√2 km/hr in a north-westerly direction. Ship D is travelling with a speed of 4√10 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. (iii) How long does it take ship D to intercept ship C?
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 – 7j 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 40^0 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.
2003 OL
The velocity of ship A is 3 i − 4 j m/s 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 α^0 north with a constant speed of 39 km/h, where tan α = 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.
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) Va = 15 i + 0 j Vb = 0 i + 20 j Vab = 15 i – 20 j Magnitude = √(15^2 + 20^2 ) = 25 m/s {we will need this later} α = tan-1^ (20/15) = 53.13^0
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 × 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:
From the diagram we can see that the shortest distance (the dotted line) corresponds to 200 sinα = 200 sin (53.13^0 ) = 160 m
2004 (b) At time t = 0, two particles P and Q are set in motion. At time t = 0, 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.
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 2
T
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.
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.
1995 (b) Two straight roads intersect at an angle of 45°. 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 220√2 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.
Two straight roads intersect at an angle 60^0. 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 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 60°. 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.
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 Vab 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 √2 km h−1. Ship B is travelling due west at 17 km h−1. (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 60° 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.
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.
A ship B is travelling in a direction 41^0 East of North at 15 m/s. A second ship C is travelling 41^0 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.
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 100m 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.
Drawing the diagram The diagram will consist of a triangle where each of the sides represents one of the following:
- the velocity of the body (Vb)
- the velocity of the carrier (Vc)
- 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.
- The carrier vector: draw this vector on its own off to the side initially.
- 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.
- 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.
- 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.
- 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).
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).
Body and Carrier exam questions 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?
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.
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.
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.
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.
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.
