6.3 Frames of reference

Frames of reference

1. A woman walks north at 3 km $·$ hr ${}^{-1}$ on a boat that is moving east at 4 km $·$ hr ${}^{-1}$ . This situation is illustrated in the diagram below.
   1. How fast is the woman moving according to her friend who is also on the boat?
   2. What is the woman's velocity according to an observer watching from the river bank?

   

2. A boy is standing inside a train that is moving at 10 m $·$ s ${}^{-1}$ to the left. The boy throws a ball in the air with a velocity of 4 m $·$ s ${}^{-1}$ . What is the resultant velocity of the ball
   1. according to the boy?
   2. according to someone outside the train?

Summary

1. Projectiles are objects that move through the air.
2. Objects that move up and down (vertical projectiles) on the earth accelerate with a constant acceleration g which is approximately equal to 9,8 m $·$ s ${}^{-2}$ directed downwards towards the centre of the earth.
3. The equations of motion can be used to solve vertical projectile problems.
   $$\begin{array}{ccc}\hfill v_f& =& v_i+gt\hfill \\ \hfill \Delta x& =& \frac{(v_i+v_f)}{2}t\hfill \\ \hfill \Delta x& =& v_{i}t+\frac{1}{2}g{t}^2\hfill \\ \hfill v_f^2& =& v_i^2+2g\Delta x\hfill \end{array}$$
4. Graphs can be drawn for vertical projectile motion and are similar to the graphs for motion at constant acceleration. If upwards is taken as positive the $\Delta x$ vs $t$ , $v$ vs $t$ ans $a$ vs $t$ graphs for an object being thrown upwards look like this:

   

5. Momentum is conserved in one and two dimensions.
   $$\begin{array}{ccc}\hfill p& =& mv\hfill \\ \hfill \Delta p& =& m\Delta v\hfill \\ \hfill \Delta p& =& F\Delta t\hfill \end{array}$$
6. An elastic collision is a collision where both momentum and kinetic energy is conserved.
   $$\begin{array}{ccc}\hfill p_{\mathrm{before}}& =& p_{\mathrm{after}}\hfill \\ \hfill K{E}_{\mathrm{before}}& =& K{E}_{\mathrm{after}}\hfill \end{array}$$
7. An inelastic collision is where momentum is conserved but kinetic energy is not conserved.
   $$\begin{array}{ccc}\hfill p_{\mathrm{before}}& =& p_{\mathrm{after}}\hfill \\ \hfill K{E}_{\mathrm{before}}& \ne & K{E}_{\mathrm{after}}\hfill \end{array}$$
8. The frame of reference is the point of view from which a system is observed.

End of chapter exercises

1. [IEB 2005/11 HG] Two friends, Ann and Lindiwe decide to race each other by swimming across a river to the other side. They swim at identical speeds relative to the water. The river has a current flowing to the east.

   

   Ann heads a little west of north so that she reaches the other side directly across from the starting point. Lindiwe heads north but is carried downstream, reaching the other side downstream of Ann. Who wins the race?
   1. Ann
   2. Lindiwe
   3. It is a dead heat
   4. One cannot decide without knowing the velocity of the current.
2. [SC 2001/11 HG1] A bullet fired vertically upwards reaches a maximum height and falls back to the ground.

   

   Which one of the following statements is true with reference to the acceleration of the bullet during its motion, if air resistance is ignored? The acceleration:
   1. is always downwards
   2. is first upwards and then downwards
   3. is first downwards and then upwards
   4. decreases first and then increases
3. [SC 2002/03 HG1] Thabo suspends a bag of tomatoes from a spring balance held vertically. The balance itself weighs 10 N and he notes that the balance reads 50 N. He then lets go of the balance and the balance and tomatoes fall freely. What would the reading be on the balance while falling?

   

   1. 50 N
   2. 40 N
   3. 10 N
   4. 0 N
4. [IEB 2002/11 HG1] Two balls, P and Q, are simultaneously thrown into the air from the same height above the ground. P is thrown vertically upwards and Q vertically downwards with the same initial speed. Which of the following is true of both balls just before they hit the ground? (Ignore any air resistance. Take downwards as the positive direction.)

   	Velocity	Acceleration
   A	The same	The same
   B	P has a greater velocity than Q	P has a negative acceleration; Q has a positive acceleration
   C	P has a greater velocity than Q	The same
   D	The same	P has a negative acceleration; Q has a positive acceleration

5. [IEB 2002/11 HG1] An observer on the ground looks up to see a bird flying overhead along a straight line on bearing 130 ${}^{\circ }$ (40 ${}^{\circ }$ S of E). There is a steady wind blowing from east to west. In the vector diagrams below, I, II and III represent the following: I       the velocity of the bird relative to the airII     the velocity of the air relative to the ground III   the resultant velocity of the bird relative to the groundWhich diagram correctly shows these three velocities?

   

6. [SC 2003/11] A ball X of mass $m$ is projected vertically upwards at a speed $u_x$ from a bridge 20 m high. A ball Y of mass $2m$ is projected vertically downwards from the same bridge at a speed of $u_y$ . The two balls reach the water at the same speed. Air friction can be ignored. Which of the following is true with reference to the speeds with which the balls are projected?
   1. $u_x=\frac{1}{2}{u}_y$
   2. $u_x=u_y$
   3. $u_x=2u_y$
   4. $u_x=4u_y$
7. [SC 2002/03 HG1] A stone falls freely from rest from a certain height. Which one of the following quantities could be represented on the $y$ -axis of the graph below?

   

   1. velocity
   2. acceleration
   3. momentum
   4. displacement
8. A man walks towards the back of a train at 2 m $·$ s ${}^{-1}$ while the train moves forward at 10 m $·$ s ${}^{-1}$ . The magnitude of the man's velocity with respect to the ground is
   1. 2 m $·$ s ${}^{-1}$
   2. 8 m $·$ s ${}^{-1}$
   3. 10 m $·$ s ${}^{-1}$
   4. 12 m $·$ s ${}^{-1}$
9. A stone is thrown vertically upwards and it returns to the ground. If friction is ignored, its acceleration as it reaches the highest point of its motion is
   1. greater than just after it left the throwers hand.
   2. less than just before it hits the ground.
   3. the same as when it left the throwers hand.
   4. less than it will be when it strikes the ground.
10. An exploding device is thrown vertically upwards. As it reaches its highest point, it explodes and breaks up into three pieces of equal mass . Which one of the following combinations is possible for the motion of the three pieces if they all move in a vertical line?

    	Mass 1	Mass 2	Mass 3
    A	v downwards	v downwards	v upwards
    B	v upwards	2v downwards	v upwards
    C	2v upwards	v downwards	v upwards
    D	v upwards	2v downwards	v downwards

11. [IEB 2004/11 HG1] A stone is thrown vertically up into the air. Which of the following graphs best shows the resultant force exerted on the stone against time while it is in the air? (Air resistance is negligible.)

    

12. What is the velocity of a ball just as it hits the ground if it is thrown upward at 10 m $·$ s ${}^{-1}$ from a height 5 meters above the ground?
13. [IEB 2005/11 HG1] A breeze of 50 km $·$ hr ${}^{-1}$ blows towards the west as a pilot flies his light plane from town A to village B. The trip from A to B takes 1 h. He then turns west, flying for $\frac{1}{2}$ h until he reaches a dam at point C. He turns over the dam and returns to town A. The diagram shows his flight plan. It is not to scale.

    

    The pilot flies at the same altitude at a constant speed of 130 km.h ${}^{-1}$ relative to the air throughout this flight.
    1. Determine the magnitude of the pilot's resultant velocity from the town A to the village B.
    2. How far is village B from town A?
    3. What is the plane's speed relative to the ground as it travels from village B to the dam at C?
    4. Determine the following, by calculation or by scale drawing:
       1. The distance from the village B to the dam C.
       2. The displacement from the dam C back home to town A.
14. A cannon (assumed to be at ground level) is fired off a flat surface at an angle, $\theta$ above the horizontal with an initial speed of $v_0$ .
    1. What is the initial horizontal component of the velocity?
    2. What is the initial vertical component of the velocity?
    3. What is the horizontal component of the velocity at the highest point of the trajectory?
    4. What is the vertical component of the velocity at that point?
    5. What is the horizontal component of the velocity when the projectile lands?
    6. What is the vertical component of the velocity when it lands?
15. [IEB 2004/11 HG1] Hailstones fall vertically on the hood of a car parked on a horizontal stretch of road. The average terminal velocity of the hailstones as they descend is 8,0 m.s ${}^{-1}$ and each has a mass of 1,2 g.
    1. Calculate the magnitude of the momentum of a hailstone just before it strikes the hood of the car.
    2. If a hailstone rebounds at 6,0 m.s ${}^{-1}$ after hitting the car's hood, what is the magnitude of its change in momentum?
    3. The hailstone is in contact with the car's hood for 0,002 s during its collision with the hood of the car. What is the magnitude of the resultant force exerted on the hood if the hailstone rebounds at 6,0 m.s ${}^{-1}$ ?
    4. A car's hood can withstand a maximum impulse of 0,48 N $·$ s without leaving a permanent dent. Calculate the minimum mass of a hailstone that will leave a dent in the hood of the car, if it falls at 8,0 m.s ${}^{-1}$ and rebounds at 6,0 m.s ${}^{-1}$ after a collision lasting 0,002 s.
16. [IEB 2003/11 HG1 - Biathlon] Andrew takes part in a biathlon race in which he first swims across a river and then cycles. The diagram below shows his points of entry and exit from the river, A and P, respectively.

    

    During the swim, Andrew maintains a constant velocity of 1,5 m.s ${}^{-1}$ East relative to the water. The water in the river flows at a constant velocity of 2,5 m.s ${}^{-1}$ in a direction 30 ${}^{\circ }$ North of East. The width of the river is 100 m. The diagram below is a velocity-vector diagram used to determine the resultant velocity of Andrew relative to the river bed.

    

    1. Which of the vectors (AB, BC and AC) refer to each of the following?
       1. The velocity of Andrew relative to the water.
       2. The velocity of the water relative to the water bed.
       3. The resultant velocity of Andrew relative to the river bed.
    2. Determine the magnitude of Andrew's velocity relative to the river bed either by calculations or by scale drawing, showing your method clearly.
    3. How long (in seconds) did it take Andrew to cross the river?
    4. At what distance along the river bank (QP) should Peter wait with Andrew's bicycle ready for the next stage of the race?
17. [IEB 2002/11 HG1 - Bouncing Ball] A ball bounces vertically on a hard surface after being thrown vertically up into the air by a boy standing on the ledge of a building.Just before the ball hits the ground for the first time, it has a velocity of magnitude 15 m.s ${}^{-1}$ . Immediately, after bouncing, it has a velocity of magnitude 10 m.s ${}^{-1}$ . The graph below shows the velocity of the ball as a function of time from the moment it is thrown upwards into the air until it reaches its maximum height after bouncing once.

    

    1. At what velocity does the boy throw the ball into the air?
    2. What can be determined by calculating the gradient of the graph during the first two seconds?
    3. Determine the gradient of the graph over the first two seconds. State its units.
    4. How far below the boy's hand does the ball hit the ground?
    5. Use an equation of motion to calculate how long it takes, from the time the ball was thrown, for the ball to reach its maximum height after bouncing.
    6. What is the position of the ball, measured from the boy's hand, when it reaches its maximum height after bouncing?
18. [IEB 2001/11 HG1] - Free Falling? A parachutist steps out of an aircraft, flying high above the ground. She falls for the first 8 seconds before opening her parachute. A graph of her velocity is shown in Graph A below.

    

    1. Use the information from the graph to calculate an approximate height of the aircraft when she stepped out of it (to the nearest 10 m).
    2. What is the magnitude of her velocity during her descent with the parachute fully open? The air resistance acting on the parachute is related to the speed at which the parachutist descends. Graph B shows the relationship between air resistance and velocity of the parachutist descending with the parachute open.

       

    3. Use Graph B to find the magnitude of the air resistance on her parachute when she was descending with the parachute open.
    4. Assume that the mass of the parachute is negligible. Calculate the mass of the parachutist showing your reasoning clearly.
19. An aeroplane travels from Cape Town and the pilot must reach Johannesburg, which is situated 1300 km from Cape Town on a bearing of 50 ${}^{\circ }$ in 5 hours. At the height at which the plane flies, a wind is blowing at 100 km $·$ hr ${}^{-1}$ on a bearing of 130 ${}^{\circ }$ for the whole trip.

    

    1. Calculate the magnitude of the average resultant velocity of the aeroplane, in km $·$ hr ${}^{-1}$ , if it is to reach its destination on time.
    2. Calculate ther average velocity, in km $·$ hr ${}^{-1}$ , in which the aeroplane should be travelling in order to reach Johannesburg in the prescribed 5 hours. Include a labelled, rough vector diagram in your answer. (If an accurate scale drawing is used, a scale of 25 km $·$ hr ${}^{-1}$ = 1 cm must be used.)
20. Niko, in the basket of a hot-air balloon, is stationary at a height of 10 m above the level from where his friend, Bongi, will throw a ball. Bongi intends throwing the ball upwards and Niko, in the basket, needs to descend (move downwards) to catch the ball at its maximum height.

    

    Bongi throws the ball upwards with a velocity of 13 m $·$ s ${}^{-1}$ . Niko starts his descent at the same instant the ball is thrown upwards, by letting air escape from the balloon, causing it to accelerate downwards. Ignore the effect of air friction on the ball.
    1. Calculate the maximum height reached by the ball.
    2. Calculate the magnitude of the minimum average acceleration the balloon must have in order for Niko to catch the ball, if it takes 1,3 s for the ball to rach its maximum height.
21. Lesedi (mass 50 kg) sits on a massless trolley. The trolley is travelling at a constant speed of 3 m $·$ s ${}^{-1}$ . His friend Zola (mass 60 kg) jumps on the trolley with a velocity of 2 m $·$ s ${}^{-1}$ . What is the final velocity of the combination (Lesedi, Zola and trolley) if Zola jumps on the trolley from
    1. the front
    2. behind
    3. the side
    (Ignore all kinds of friction)