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4.4 Mechanics and Materials - Projectile motion - Qs

Q1. An electric wheelchair, powered by a battery, allows the user to move around independently.

One type of electric wheelchair has a mass of 55 kg. The maximum distance it can travel on level ground is 12 km when carrying a user of mass 65 kg and travelling at its maximum speed of 1.5 m s−1.

The battery used has an emf of 12 V and can deliver 7.2 × 104 C as it discharges fully.

(a) Show that the average power output of the battery during the journey is about 100 W.

(3)

(b) During the journey, forces due to friction and air resistance act on the wheelchair and its user.

Assume that all the energy available in the battery is used to move the wheelchair and its user during the journey.

Calculate the total mean resistive force that acts on the wheelchair and its user.

total mean resistive force = ____________________ N (2)

The diagram below shows the wheelchair and its user travelling up a hill. The hill makes an angle of 4.5° to the horizontal.

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(c) Calculate the force that gravity exerts on the wheelchair and its user parallel to the slope.

force parallel to the slope = ____________________ N (1)

(d) Calculate the maximum speed of the wheelchair and its user when travelling up this hill when the power output of the battery is 100 W.

Assume that the resistive forces due to friction and air resistance are the same as in part (b).

maximum speed = ____________________ m s−1

(2)

(e) Explain how and why the maximum range of the wheelchair on level ground is affected by

• the mass of the user

• the speed at which the wheelchair travels.

Effect of mass _______________________________________________________

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Effect of speed ______________________________________________________

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(4) (Total 12 marks)

Q2. (a) Describe how a beam of fast moving electrons is produced in the cathode ray tube of

an oscilloscope.

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___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(b) The figure below shows the cathode ray tube of an oscilloscope. The details of how the beam of electrons is produced are not shown.

The electron beam passes between two horizontal metal plates and goes on to strike a fluorescent screen at the end of the tube. The plates are 0.040 m long and are separated by a gap of 0.015 m. A potential difference of 270 V is maintained between the plates.An individual electron takes 1.5 × 10–9 s to pass between the plates.The distance between the right-hand edge of the plates and the fluorescent screen is 0.20 m.

(i) Show that the vertical acceleration of an electron as it passes between the horizontal metal plates is approximately 3.2 × 1015 ms–2.

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(3)

(ii) Show that the vertical distance travelled by an electron as it passes between the horizontal metal plates is approximately 3.6 mm.

(2)

(iii) Show that the vertical component of velocity achieved by an electron in the beam by the time it reaches the end of the plates is approximately 4.7 × 106 m s–1.

(2)

(iv) Calculate the vertical displacement, y, of the electron beam from the centre of the screen. Give your answer in m.

vertical displacement ____________________ m (3)

(Total 13 marks)

Q3.

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A projectile is launched some distance above the ground at an angle of 25° above the horizontal with a vertical component of velocity of 5.0 m s−1. Figure 1 shows the flight path of the projectile. The flight takes 1.3 s.

Ignore the effects of air resistance throughout this question.

Figure 1

(a) (i) Show that the initial speed of the projectile is about 12 m s−1.

(2)

(ii) Calculate the horizontal component of velocity as the projectile hits the ground.

horizontal component of velocity = ____________________ m s−1

(2)

(b) (i) Calculate the maximum height above the starting point reached by the projectile. Give your answer to an appropriate number of significant figures.

maximum height reached = ____________________ m (2)

(ii) Calculate the total horizontal distance travelled by the projectile from its starting point.

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horizontal distance = ____________________ m (1)

(c) (i) Mark with an A on the flight path in Figure 1 the position where the speed of the projectile is greatest.

(1)

(ii) Mark with a B on the flight path in Figure 1 the position where the speed of the projectile is least.

(1)

(iii) The projectile reaches its maximum height at time tH and finishes its flight at time tF. Draw on Figure 2 a graph to show how the magnitude of the vertical component of velocity of the projectile varies with time. Numerical values are not required.

Figure 2

(2)

(Total 11 marks)

Q4. The diagram below shows two different rifles being fired horizontally from a height of 1.5 m above ground level. Assume the air resistance experienced by the bullets is negligible.

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(a) When rifle A is fired, the bullet has a horizontal velocity of 430 m s–1 as it leaves the rifle. Assume the ground is level.

(i) Calculate the time that the bullet is in the air before it hits the ground.

time ____________________ s (2)

(ii) Calculate the horizontal distance travelled by the bullet before it hits the ground.

horizontal distance ____________________ m (1)

(b) Rifle B is fired and the bullet emerges with a smaller horizontal velocity than the bullet from rifle A.

Explain why the horizontal distance travelled by bullet B will be less than bullet A.

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___________________________________________________________________

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___________________________________________________________________ (3)

(Total 6 marks)

Q5. The motion of a long jumper during a jump is similar to that of a projectile moving under gravity. The figure below shows the path of an athlete above the ground during a long jump from half-way through the jump at position A, to position B at which contact is made with sand on the ground. The athlete is travelling horizontally at A.

(a) During this part of the jump, the centre of mass of the athlete falls 1.2 m.

(i) Calculate the time between positions A and B.

time ____________________ s (3)

(ii) The athlete is moving horizontally at A with a velocity of 8.5 m s–1. Assume there is noair resistance. Calculate the horizontal displacement of the centre of mass from A to B.

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horizontal displacement ____________________ m (2)

(b) (i) The athlete in the image above slides horizontally through the sand a distance of 0.35 m before stopping.

Calculate the time taken for the athlete to stop. Assume the horizontal component of the resistive force from the sand is constant.

time ____________________ s (2)

(ii) The athlete has a mass of 75 kg. Calculate the horizontal component of the resistive force from the sand.

horizontal component of resistive force ____________________ N (3)

(Total 10 marks)

Q6. In the cathode ray tube illustrated below, electrons are accelerated by a potential difference of 1.8 kV between the cathode (C) and the anode (A).

(a) (i) Calculate the kinetic energy, in J, of the electrons after they have passed the

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anode.

charge on an electron, e = –1.6 × 10–19 C (2)

(ii) Calculate the velocity of the electrons after they have passed the anode.

Mass of an electron = 9.1 × 10–31 kg (2)

(b) The plates P and Q are 8.0 cm long and are separated by a gap of 4.0 cm.

(i) Define electric field strength.

______________________________________________________________

______________________________________________________________ (1)

(ii) Calculate the force acting on an electron when it is between P and Q and state the direction of the force.

Direction _______________________________________________________

(3)

(iii) Calculate the time taken for an electron to pass between the plates. (1)

(iv) Calculate the vertical component of velocity at the time the electron leaves the electric field between P and Q.

(2)

(v) Calculate the additional vertical displacement of the electron between the time it leaves the electric field between P and Q and when it reaches the screen.

(1) (Total 12 marks)

Q7. (a) Explain why a particle is accelerating even when it is moving with a uniform speed

in a circular path.

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___________________________________________________________________

___________________________________________________________________ (2)

(b) Figure 1 shows a schematic diagram of a proton synchrotron. This is a device for accelerating protons to high speeds in a horizontal circular path.

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Figure 1

In the synchrotron the protons of mass 1.7 × 10–27 kg are injected at point A at a speed of 8.0 × 106 m s–1. The diameter of the path taken by the protons is 400 m.

(i) Show on Figure 1 the direction of the force required to make a proton move in the circular path when the proton is at the position marked P.

(1)

(ii) Calculate the force that has to be provided to produce the circular path when the speed of a proton is 8.0 × 106 m s–1.

(2)

(iii) Sketch on Figure 2 a graph to show how this force will have to change as the speed of the proton increases over the range shown on the x-axis. Insert an appropriate scale on the force axis.

Figure 2

(c) Before reaching their final energy the protons in the synchrotron in part (b) travel around the accelerator 420 000 times in 2.0 s.

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acceleration of free fall, g = 9.8 m s–2

(i) Calculate the total distance travelled by a proton in the 2.0 s time interval. (2)

(ii) Unless a vertical force is applied the protons wold fall as they move through the horizontal channel.

Calculate the distance a proton would fall in two seconds. (2)

(iii) Determine the force necessary to prevent the vertical movement. (1)

(Total 12 marks)

Q8. (a) The diagram below shows part of a precipitation system used to collect dust

particles in a chimney. It consists of two large parallel vertical plates maintained at potentials of +25 kV and –25 kV.

The diagram below also shows the electric field lines between the plates.

(i) Add arrows to the diagram to show the direction of the electric field. (1)

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(ii) Explain what is meant by an equipotential surface.

______________________________________________________________

______________________________________________________________

______________________________________________________________ (1)

(iii) Draw and label on the diagram equipotentials that correspond to potentials of –12.5 kV, 0 V, and +12.5 kV.

(2)

(b) A small dust particle moves vertically up the centre of the chimney, midway between the plates.

(i) The charge on the dust particle is +5.5 nC. Show that there is an electrostatic force on the particle of about 0.07 mN.

(2)

(ii) The mass of the dust particle is 1.2 × 10–4 kg and it moves up the centre of the chimney at a constant vertical speed of 0.80 m s–1.

Calculate the minimum length of the plates necessary for this particle to strike one of them. Ignore air resistance.

(4) (Total 10 marks)

Q9. The graph shows how the position of a steel ball which has been projected horizontally from P changes with time. The position of the ball is shown at constant time intervals.

(a) Explain how the horizontal motion of the ball shows that air resistance is negligible.

___________________________________________________________________

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___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(b) Explain the vertical motion of the ball.

___________________________________________________________________

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___________________________________________________________________ (2)

(c) If air resistance were not negligible, describe how this would affect

(i) the horizontal motion of the ball,

______________________________________________________________

______________________________________________________________

(ii) the vertical motion of the ball.

______________________________________________________________

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______________________________________________________________ (3)

(Total 7 marks)

Q10. A ball is dropped and rebounds vertically to less than the original height.

For this first bounce only, sketch graphs of

(a) the velocity of the ball plotted against time,

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(4)

(b) the acceleration of the ball plotted against time.

(1)

(c)

The ball is then thrown at an angle to the horizontal and follows the trajectory shown in the diagram.

Mark on the diagram the directions of

(i) the acceleration vector at P,

(ii) the acceleration vector at Q,

(iii) the momentum vector at P,

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(iv) the momentum vector at Q. (4)

(d) The mass of the ball is 0.15 kg and the initial direction makes an angle of 50° to the horizontal. Calculate the magnitude of the momentum of the ball at Q when it is projected with an initial speed of 15 m s–1. Neglect the effects of air resistance.

___________________________________________________________________

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___________________________________________________________________ (4)

(Total 13 marks)

Q11. Figure 1 shows a kite boarder holding a line that is attached to a kite.

Figure 1

The wind blows the kite and the kite boarder moves at a constant speed across a level water surface. The tension in the line is 720 N and the line makes an angle of 50° to the horizontal.

(a) (i) Calculate the vertical component of the tension in the line.

vertical component of tension ____________________ N (2)

(ii) The kite boarder has a mass of 58 kg.

Calculate the normal reaction of the board on the kite boarder.

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normal reaction ____________________ N (2)

(iii) Suggest how the answer to part (a)(ii) compares with the upthrust of the water on the board.

Consider the board to have negligible mass.

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______________________________________________________________ (2)

(b) Figure 2 shows the kite boarder about to perform a jump using a ramp.

Figure 2

The end of the ramp is 1.8 m above the water surface. The kite boarder leaves the ramp at a velocity of 12 m s−1 and at an angle of 17° to the horizontal. The kite boarder lets go of the line at the instant he leaves the ramp.

Calculate the speed with which the kite boarder enters the water.

Assume that the kite boarder is a point mass and ignore the effects of air resistance.

speed ____________________ m s−1

(4) (Total 10 marks)

Q12. While investigating projectile motion, a student used stroboscopic photography to determine the position of a steel ball at regular intervals as it fell under gravity. With the stroboscope flashing 20 times per second, the ball was released from rest at the top of an

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inclined track, and left the foot of the track at P, as shown in the diagram below.

For each of the images on the photograph, the student calculated the horizontal distance, x, and the vertical distance, y, covered by the ball at time t after passing P. Both distances were measured from point P. He recorded his results for the distances x and y in the table.

image x/cm y/cm t/s (y/t)/cm s–1

1 11.6 9.3 0.05

2 22.0 21.0 0.10

3 32.4 35.0 0.15

4 44.2 51.8 0.20

5 54.8 71.0 0.25

6 66.0 92.2 0.30

(a) Using two sets of measurements from the table, calculate the horizontal component of velocity of the ball. Give a reason for your choice of measurements.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(b) The student worked out that the variables y and t in the experiment could be represented by

= u + kt

where u and k are constants.

(i) Complete the table above.

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(ii) Use the data in the table to plot a suitable graph to confirm the equation.

(Allow one sheet pf graph paper)

(iii) Use your graph to find the values of u and k.

______________________________________________________________

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______________________________________________________________ (9)

(c) State the physical significance of

u _________________________________________________________________

___________________________________________________________________

k _________________________________________________________________

___________________________________________________________________ (2)

(d) Calculate the magnitude of the velocity of the ball at point P.

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___________________________________________________________________ (2)

(Total 15 marks)

Q13. The aeroplane shown in the diagram below is travelling horizontally at 95 m s–1. It has to drop a crate of emergency supplies. The air resistance acting on the crate may be neglected.

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(a) (i) The crate is released from the aircraft at point P and lands at point Q. Sketch the path followed by the crate between P and Q as seen from the ground.

(ii) Explain why the horizontal component of the crate’s velocity remains constant while it is moving through the air.

______________________________________________________________

______________________________________________________________

______________________________________________________________ (3)

(b) (i) To avoid damage to the crate, the maximum vertical component of the crate’s velocity on landing should be 32 m s–1. Show that the maximum height from which the crate can be dropped is approximately 52 m.

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(ii) Calculate the time taken for the crate to reach the ground if the crate is dropped from a height of 52 m.

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(iii) If R is a point on the ground directly below P, calculate the horizontal distance QR.

______________________________________________________________

______________________________________________________________ (6)

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(c) In practice air resistance is not negligible. State and explain the effect this has on the maximum height from which the crate can be dropped.

___________________________________________________________________

___________________________________________________________________ (2)

(Total 11 marks)

Q14. The figure below shows a skateboarder descending a ramp.

The skateboarder starts from rest at the top of the ramp at A and leaves the ramp at B horizontally with a velocity v.

(a) State the energy changes that take place as the skateboarder moves from A to B.

___________________________________________________________________

___________________________________________________________________ (2)

(b) In going from A to B the skateboarder’s centre of gravity descends a vertical height of 1.5 m. Calculate the horizontal velocity, v, stating an assumption that you make.

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___________________________________________________________________ (3)

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(c) Explain why the acceleration decreases as the skateboarder moves from A to B.

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___________________________________________________________________ (2)

(d) After leaving the ramp at B the skateboarder lands on the ground at C 0.42 s later.

Calculate for the skateboarder

(i) the horizontal distance travelled between B and C,

______________________________________________________________

______________________________________________________________

(ii) the vertical component of the velocity immediately before impact at C,

______________________________________________________________

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(iii) the magnitude of the resultant velocity immediately before impact at C.

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______________________________________________________________ (5)

(Total 12 marks)

Q15. English longbows from the time of Henry VIII were found on board a sunken Tudor warship. The diagram below shows how the horizontal force, F, exerted by an archer on the bow string varies with the horizontal displacement, d, of the arrow as the bow is drawn.

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(a) (i) Show that the energy stored in the bow is

Fmaxdmax

where Fmax is the maximum force in the bow and dmax is the maximum displacement of the bow string.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(ii) Show that the maximum speed, v, with which an arrow can leave the bow is given by

v =

where m is the mass of the arrow.

______________________________________________________________

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______________________________________________________________

______________________________________________________________ (2)

(b) Each arrow had a mass of 0.060 kg and the bow string was displaced by a maximum of 0.60 m before releasing the arrow. The archers needed to exert a maximum force of 650 N.

(i) Show that the maximum speed of the arrow as it leaves the bow is about 80 m s–1.

______________________________________________________________

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______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(ii) State one reason why the speed of the arrow as it leaves the bow is likely to be less than 80 m s–1.

______________________________________________________________

______________________________________________________________ (1)

(c) The initial speed of an arrow as it leaves a bow is 65 m s–1. When leaving the bow the arrow travels at an angle of 55° to the horizontal. Assume that the air resistance is negligible.

(i) Show that the vertical component of the initial speed of the arrow is about 53 m s–1.

______________________________________________________________

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______________________________________________________________ (1)

(ii) The arrow is fired from ground level across a flat, level field. Show that the arrow is in flight for about 11 s.

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______________________________________________________________ (3)

(iii) Calculate the horizontal distance travelled by the arrow when fired from level ground.

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______________________________________________________________

horizontal distance travelled ____________________ m (2)

(Total 13 marks)

Q16. An electric shower heats the water flowing through it from 10°C to 42°C when the volume flow rate is 5.2 × 10−5 m3 s−1.

(a) (i) Calculate the mass of water flowing through the shower each second.

density of water = 1000 kg m−3

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) Calculate the power supplied to the shower, assuming all the electrical energy supplied to it is gained by the water as thermal energy.

specific heat capacity of water = 4200 J kg−1 K−1.

______________________________________________________________

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______________________________________________________________

______________________________________________________________ (4)

(b) A jet of water emerges horizontally at a speed of 2.5 m s−1 from a hole in the shower head. The hole is 2.0 m above the floor of the shower. Calculate the horizontal distance travelled by this jet. Assume air resistance is negligible.

___________________________________________________________________

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___________________________________________________________________ (3)

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(Total 7 marks)

Q17. Figure 1 shows the flight of a cricket ball hit by a batsman at 30° to the horizontal at a speed of 22 m s–1. The ball reached a fielder without bouncing and was caught at the same height as it was hit. The effect of air resistance on the cricket ball is negligible.

Figure 1

(a) (i) Calculate the vertical speed of the ball at the instant it left the bat.

vertical speed ____________________ m s–1

(1)

(ii) Show that the ball was in the air for about 2.2 s.

(3)

(iii) How far did the ball travel horizontally before it was caught?

distance ____________________ m (1)

(b) (i) A tennis ball is about the same size as a cricket ball but has a lower mass. By considering the energy changes that take place, explain why a tennis ball hit at the same speed and angle as the cricket ball would be unlikely to reach the fielder without bouncing.

______________________________________________________________

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______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (3)

(ii) Draw on Figure 2 the path you would expect a tennis ball to follow when hit at the same speed and angle as the cricket ball.

Figure 2

(2)

(Total 10 marks)

Q18. A digital camera was used to obtain a sequence of images of a tennis ball being struck by a tennis racket. The camera was set to take an image every 5.0 ms. The successive positions of the racket and ball are shown in the diagram below.

(a) The ball has a horizontal velocity of zero at A and reaches a constant horizontal velocity at D as it leaves the racket. The ball travels a horizontal distance of 0.68 m between D and G.

(i) Show that the horizontal velocity of the ball between positions D and G in the diagram above is about 45 m s–1.

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(3)

(ii) Calculate the horizontal acceleration of the ball between A and D.

answer = ____________________ m s–2

(1)

(b) At D, the ball was projected horizontally from a height of 2.3 m above level ground.

(i) Show that the ball would fall to the ground in about 0.7 s.

(3)

(ii) Calculate the horizontal distance that the ball will travel after it leaves the racket before hitting the ground. Assume that only gravity acts on the ball as it falls.

answer = ____________________ m (2)

(iii) Explain why, in practice, the ball will not travel this far before hitting the ground.

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(Total 11 marks)

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Q19. The diagram below shows the path of a ball thrown horizontally from the top of a tower of height 24 m which is surrounded by level ground.

(a) Using two labelled arrows, show on the diagram above the direction of the velocity, v, and the acceleration, a, of the ball when it is at point P.

(2)

(b) (i) Calculate the time taken from when the ball is thrown to when it first hits the ground. Assume air resistance is negligible.

Answer ____________________ s (2)

(ii) The ball hits the ground 27 m from the base of the tower. Calculate the speed at which the ball is thrown.

Answer ____________________ m s–1

(2) (Total 6 marks)

Q20. The figure below shows the path that a tennis ball would follow in the absence of air resistance, after being hit horizontally at A.

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(a) Explain why the path of the ball is curved in this way.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(b) Draw onto the figure the path of a ball, hit in the same way at A, that is affected by air resistance.

(1) (Total 3 marks)

Q21. A tennis player serves a ball from a height of 2.51 m at 18.0 ms–1 in a horizontal direction. The ball just clears the net which is 1.00 m high. In this question assume that air resistance is negligible.

The figure below shows the ball and its resulting trajectory across the court.

(a) Show that the ball takes approximately 0.6 s to reach the net after being served.

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(3)

(b) (i) Calculate the vertical component of the velocity of the ball as it passes over the net.

vertical component of velocity ____________________ ms–1

(2)

(ii) Calculate the overall velocity of the ball as it passes over the net.

magnitude of velocity ____________________ ms–1

angle to horizontal ____________________ degree (3)

(Total 8 marks)

Q22. In a castle, overlooking a river, a cannon was once employed to fire at enemy ships. One ship was hit by a cannonball at a horizontal distance of 150 m from the cannon as shown in the figure below. The height of the cannon above the river was 67 m and the cannonball was fired horizontally.

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(a) (i) Show that the time taken for the cannonball to reach the water surface after being fired from the cannon was 3.7 s. Assume the air resistance was negligible.

(2)

(ii) Calculate the velocity at which the cannonball was fired. Give your answer to an appropriate number of significant figures.

answer = ______________________ m s–1

(2)

(iii) Calculate the vertical component of velocity just before the cannonball hit the ship.

answer = ______________________ m s–1

(2)

(iv) By calculation or scale drawing, find the magnitude and direction of the velocity of the cannonball just before it hit the ship.

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velocity = ______________________ m s–1

direction = ______________________ (4)

(b) (i) Calculate the loss in gravitational potential energy of the cannonball. mass of the cannonball = 22 kg

answer = ______________________ J (1)

(ii) Describe the energy changes that take place from the moment the cannonball leaves the cannon until just before it hits the water. Include the effects of air resistance.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(Total 13 marks)

Q23. The figure below shows a system that separates two minerals from the ore containing them using an electric field.

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The crushed particles of the two different minerals gain opposite charges due to friction as they travel along the conveyor belt and through the hopper. When they leave the hopper they fall 4.5 metres between two parallel plates that are separated by 0.35 m.

(a) Assume that a particle has zero velocity when it leaves the hopper and enters the region between the plates.

Calculate the time taken for this particle to fall between the plates.

time taken = _____________s (2)

(b) A potential difference (pd) of 65 kV is applied between the plates.

Show that when a particle of specific charge 1.2 × 10–6 C kg–1 is between the plates its horizontal acceleration is about 0.2 m s–2.

(3)

(c) Calculate the total horizontal deflection of the particle that occurs when falling between the plates.

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horizontal deflection = _____________m (1)

(d) Explain why the time to fall vertically between the plates is independent of the mass of a particle.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(e) State and explain two reasons, why the horizontal acceleration of a particle is different for each particle.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (4)

(Total 12 marks)

Q24. A car is designed to break the land speed record. The thrust exerted on the car is 230 kN at one instant of its motion. The mass of the car at this instant is 11 000 kg.

(a) The acceleration of the car at this instant is 2.9 m s−2.

Calculate the air resistance acting on the car.

air resistance =__________________ N (3)

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(b) The thrust on the car remains constant as the speed increases.

Explain why the acceleration decreases and eventually reaches zero.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(c) A supersonic car is attempting to break the land speed record on a horizontal track. When it is travelling at 320 m s−1, a small part P that is 1.5 m above the ground becomes detached from the car. The initial vertical velocity of P is 2.5 m s−1 in the upwards direction.

Calculate the time taken for the small part P to reach the ground. Assume that air resistance has a negligible effect on the vertical motion.

time =______________________s (3)

(d) The graph below shows the path that P would follow from the instant that it became detached if there were no air resistance in the horizontal direction.

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In practice, air resistance is not negligible in the horizontal direction.

Draw, on the graph, a line to show the path that P would follow assuming that air resistance only affects motion in the horizontal direction.

(2)

(e) Explain your answer to part (d), including the reason why air resistance is negligible in the vertical direction.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 12 marks)

Q25. Figure 1 shows a golfer hitting a ball from the top of a cliff. The ball follows the path shown. The ball is hit with an initial velocity of 40 m s−1 at an angle of 30° above the horizontal, as shown. Assume that there is no air resistance.

Figure 1

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(a) Calculate the initial vertical component of velocity of the ball.

initial vertical component of velocity = ______________________________ m s−1

(1)

(b) Draw on the diagram an arrow to show the direction of the force acting on the ball when it is at point X, the highest point of the flight. Label this arrow F.

(1)

(c) At point Y the ball is level with its initial position.

Show that the time taken to reach Y is about 4 s. (2)

(d) The total time of flight of the ball is 6.0 s.

Show on Figure 2 how v, the vertical component of the velocity, changes throughout the whole 6.0 s.

Figure 2

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(3)

(e) Calculate the height h of the cliff.

height _________________ m (3)

(f) In practice, the air resistance affects the path of the ball.

Draw on Figure 1 the path the ball takes when air resistance is taken into account. (2)

(Total 12 marks)

Q26. (a) Figure 1 shows a skier descending the ramp of a ski jump. Figure 2 shows a graph

of the distance travelled along the ramp against time, from the time the descent starts until the skier leaves the end of the ramp.

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Figure 1

Figure 2

The skier of mass 80 kg (including equipment) skis down the ramp and leaves it horizontally. The skier gains 55% of the available gravitational potential energy as kinetic energy when descending the ramp.

acceleration of free fall, g = 9.8 m s–2

(i) One energy transformation which occurs as the skier skis down the ramp is from gravitational potential energy to kinetic energy of the skier. State two other energy transformations that occur as the skier skis down the ramp.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(ii) Use Figure 2 to show that the speed at which the skier leaves the ramp is about 23 m s–1. Show your reasoning clearly.

(2)

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(iii) Determine the height of the ramp. (3)

(b) Figure 1 shows the path taken by the skier after leaving the ramp. Assuming that there was no lift or drag due to the air during this jump, calculate:

(i) the time for which the skier was in flight; (2)

(ii) the horizontal distance jumped by the skier before landing. (2)

(c) On landing the skier has considerable vertical momemtum that has to be reduced to zero. The surface on which the skier lands is hard-packed snow. To reduce the force experienced by the skier, the landing surface is angled at 40° to the horizontal.

Explain briefly how angling the landing surface reduces the vertical component of the force, experienced by the skier.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(Total 14 marks)

Q27. A rugby ball is kicked towards the goal posts shown in the diagram below from a position directly in front of the posts. The ball passes over the cross-bar and between the posts.

(a) The ball takes 1.5 s to reach a point vertically above the cross-bar of the posts.

(i) Calculate the ball's horizontal component of velocity, vh. Ignore air resistance.

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vh ____________________ (2)

(ii) The ball reaches its maximum height at the same time as it passes over the crossbar. State the vertical component of velocity when the ball is at its maximum height.

______________________________________________________________ (1)

(iii) The ball’s maximum height is 11 m. Calculate, vv, the vertical component of velocity of the ball immediately after it has been kicked. Ignore the effects of air resistance.

acceleration due to gravity, g = 9.8 m s–2

vv ____________________ (3)

(b) (i) Determine the magnitude of the initial velocity, v, of the ball immediately after it is kicked.

v ____________________ (3)

(ii) Determine the angle above the horizontal at which the ball was kicked.

Angle ____________________ (1)

(c) State and explain at what instant the ball will have its maximum kinetic energy.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 12 marks)

Q28. A golf ball was hit from the surface of the Moon. The time of flight was 4.0 s.

What is the best estimate for the maximum height reached by the ball?

acceleration due to gravity on the Moon = 1.6 m s–2

A 3 m

B 15 m

C 40 m

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D 80 m (Total 1 mark)

Q29. Two identical balls, X and Y, are at the same height and a horizontal distance of 25 cm apart.

X is projected horizontally with a velocity of 0.10 m s–1 towards Y at the same time that Y is released from rest. Both X and Y move freely in the absence of air resistance.

What is the distance between the balls 1.0 s later?

A 0.15 m

B 0.25 m

C 2.4 m

D 4.9 m (Total 1 mark)

Q30. Which line, A to D, in the table correctly describes the trajectory of charged particles which enter separately, at right angles, a uniform electric field, and a uniform magnetic field?

uniform electric field uniform magnetic field

A parabolic circular

B circular parabolic

C circular circular

D parabolic parabolic

(Total 1 mark)

Q31. A ballbearing X of mass 2m is projected vertically upwards with speed u. A ballbearing Y of mass m is projected at 30° to the horizontal with speed 2u at the same time. Air resistance is negligible. Which of the following statements is correct?

A The horizontal component of Y's velocity is u.

B The maximum height reached by Y is half that reached by X

C X and Y reach the ground at the same time.

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D X reaches the ground first. (Total 1 mark)

Q32. A beam of electrons, moving with a constant velocity v in a vacuum, enters a uniform electric field between two metal plates.

Which line, A to D, in the table describes the components of the acceleration of the electrons in the x and y directions as they move through the field?

acceleration in x direction acceleration in y direction

A zero zero

B zero constant

C constant zero

D constant constant

(Total 1 mark)

Q33. A ball of mass 0.20 kg is thrown and moves in a curved path, as shown below. At Q it is travelling horizontally.

Assume air resistance is negligible.

What is the momentum of the ball at Q?

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A zero

B 2.0 N s

C 3.5 N s

D 4.0 N s (Total 1 mark)

Q34. The diagram shows the path of a projectile launched from ground level with a speed of 25 m s–1 at an angle of 42° to the horizontal.

What is the horizontal distance from the starting point of the projectile when it hits the ground?

A 23 m

B 32 m

C 47 m

D 63 m (Total 1 mark)

Q35. A stone of mass 0.4 kg is projected horizontally at a speed of 6.0 m s−1 from the top of a wall, 5.0 m above the surrounding ground. When it arrives at the ground its speed is 10 m s−1.

How much energy is lost by the stone in falling through the air?

A 2.4 J

B 6.8 J

C 12.8 J

D 14.4 J (Total 1 mark)