3.02d Constant acceleration: SUVAT formulae

2 A lift is travelling upwards and accelerating uniformly. During a 5 second period, it travels 16 metres and the speed of the lift increases from \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. \(\quad\) Find \(u\).
2. Find the acceleration of the lift.

5 A block, of mass 14 kg , is held at rest on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.25 . A light inextensible string, which passes over a fixed smooth peg, is attached to the block. The other end of the string is attached to a particle, of mass 6 kg , which is hanging at rest. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-10_264_716_502_708} The block is released and begins to accelerate.

1. Find the magnitude of the friction force acting on the block.
2. By forming two equations of motion, one for the block and one for the particle, show that the magnitude of the acceleration of the block and the particle is \(1.225 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.
4. When the block is released, it is 0.8 metres from the peg. Find the speed of the block when it hits the peg.
5. When the block reaches the peg, the string breaks and the particle falls a further 0.5 metres to the ground. Find the speed of the particle when it hits the ground.
   (3 marks)
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-11_2484_1709_223_153}

6 A ball is kicked from the point \(P\) on a horizontal surface. It leaves the surface with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal and hits the surface for the first time at the point \(Q\). Assume that the ball is a particle that moves only under the influence of gravity. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-12_317_1118_513_461}

1. Show that the time that it takes the ball to travel from \(P\) to \(Q\) is 3.13 s , correct to three significant figures.
2. Find the distance between the points \(P\) and \(Q\).
3. If a heavier ball were projected from \(P\) with the same velocity, how would the distance between \(P\) and \(Q\), calculated using the same modelling assumptions, compare with your answer to part (b)? Give a reason for your answer.
4. Find the maximum height of the ball above the horizontal surface.
5. State the magnitude and direction of the velocity of the ball as it hits the surface.
   
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-13_2484_1709_223_153}

6 Two particles, \(A\) and \(B\), have masses 12 kg and 8 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg, as shown in the diagram. $$A ( 12 \mathrm {~kg} )$$ The particles are released from rest and move vertically. Assume that there is no air resistance.

1. By forming two equations of motion, show that the magnitude of the acceleration of each particle is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. After the particles have been moving for 2 seconds, both particles are at a height of 4 metres above a horizontal surface. When the particles are in this position, the string breaks.
   1. Find the speed of particle \(A\) when the string breaks.
   2. Find the speed of particle \(A\) when it hits the surface.
   3. Find the time that it takes for particle \(B\) to reach the surface after the string breaks. Assume that particle \(B\) does not hit the peg.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-13_2484_1709_223_153}

8 A ball is struck so that it leaves a horizontal surface travelling at \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-16_293_1364_461_347}

1. Show that the ball takes \(\frac { 3 \sin \alpha } { 2 }\) seconds to reach its maximum height.
2. The ball reaches a maximum height of 7 metres.
   1. Find \(\alpha\).
   2. Find the range, \(O A\).
3. State two assumptions that you needed to make in order to answer the earlier parts of this question. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-17_2347_1691_223_153} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-18_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-19_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-20_2505_1734_212_138}

1 A crane is used to lift a load, using a single vertical cable which is attached to the load. The load accelerates uniformly from rest. When it has risen 0.9 metres, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. 1. Show that the acceleration of the load is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the time taken for the load to rise 0.9 metres.
2. Given that the mass of the load is 800 kg , find the tension in the cable while the load is accelerating.

3 A pair of cameras records the time that it takes a car on a motorway to travel a distance of 2000 metres. A car passes the first camera whilst travelling at \(32 \mathrm {~ms} ^ { - 1 }\). The car continues at this speed for 12.5 seconds and then decelerates uniformly until it passes the second camera when its speed has decreased to \(18 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the distance travelled by the car in the first 12.5 seconds.
2. Find the time for which the car is decelerating.
3. Sketch a speed-time graph for the car on this 2000-metre stretch of motorway.
4. Find the average speed of the car on this 2000-metre stretch of motorway.

5 Two particles, \(P\) and \(Q\), are connected by a string that passes over a fixed smooth peg, as shown in the diagram. The mass of \(P\) is 5 kg and the mass of \(Q\) is 3 kg . \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-3_209_433_1009_808} The particles are released from rest in the position shown.

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. State two modelling assumptions that you have made about the string.
4. Particle \(P\) hits the floor when it has moved 0.196 metres and \(Q\) has not reached the peg.
   1. Find the time that it takes \(P\) to reach the floor.
   2. Find the speed of \(P\) when it hits the floor.

6 A bullet is fired horizontally from the top of a vertical cliff, at a height of \(h\) metres above the sea. It hits the sea 4 seconds after being fired, at a distance of 1000 metres from the base of the cliff, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-4_353_901_479_571}

1. Find the initial speed of the bullet.
2. \(\quad\) Find \(h\).
3. Find the speed of the bullet when it hits the sea.
4. Find the angle between the velocity of the bullet and the horizontal when it hits the sea.

3 A car is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The driver applies the brakes and a constant braking force acts on the car until it comes to rest.

1. Assume that no other horizontal forces act on the car.
   1. After the car has travelled 75 metres, its speed has reduced to \(10 \mathrm {~ms} ^ { - 1 }\). Find the acceleration of the car.
   2. Find the time taken for the speed of the car to reduce from \(20 \mathrm {~ms} ^ { - 1 }\) to zero.
   3. Given that the mass of the car is 1400 kg , find the magnitude of the constant braking force.
2. Given that a constant air resistance force of magnitude 200 N acts on the car during the motion, find the magnitude of the constant braking force.
   (1 mark)

5 Two particles are connected by a light inextensible string that passes over a smooth peg. The particles have masses of 3 kg and 1 kg . The 1 kg particle is pulled down to ground level, where it is 40 cm below the level of the 3 kg particle, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_490_648_1272_696} The particles are released from rest with the string vertical above each particle. Assume that no resistance forces act on the particles as they move.

1. By forming two equations of motion, one for each particle, find the magnitude of the acceleration of the particles after they have been released but before the 3 kg particle hits the ground.
2. Find the speed of the 1 kg particle when the 3 kg particle hits the ground.
3. After the 3 kg particle has hit the ground, the 1 kg particle continues to move and the string is now slack. Find the maximum height above ground level reached by the 1 kg particle.
4. If a constant air resistance force also acts on the particles as they move, explain how this would change your answer for the acceleration in part (a). Give a reason for your answer.

1 A car is travelling along a straight horizontal road. It is moving at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it starts to accelerate. It accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 12 seconds.

1. Find the speed of the car at the end of the 12 seconds.
2. Find the distance travelled during the 12 seconds.
3. The mass of the car is 1400 kg . A horizontal forward driving force of 1600 N acts on the car during the 12 seconds. Find the magnitude of the resistance force that acts on the car.
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-02_1513_1709_1192_153}

3 A skip, of mass 800 kg , is at rest on a rough horizontal surface. The coefficient of friction between the skip and the ground is 0.4 . A rope is attached to the skip and then the rope is pulled by a van so that the rope is horizontal while it is taut, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_237_1118_497_463} The mass of the van is 1700 kg . A constant horizontal forward driving force of magnitude \(P\) newtons acts on the van. The skip and the van accelerate at \(0.05 \mathrm {~ms} ^ { - 2 }\). Model both the van and the skip as particles connected by a light inextensible rope. Assume that there is no air resistance acting on the skip or on the van.

1. Find the speed of the van and the skip when they have moved 6 metres.
2. Draw a diagram to show the forces acting on the skip while it is accelerating.
3. Draw a diagram to show the forces acting on the van while it is accelerating. State one advantage of modelling the van as a particle when considering the vertical forces.
4. Find the magnitude of the friction force acting on the skip.
5. Find the tension in the rope.
6. \(\quad\) Find \(P\).
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_771_1703_1932_155}

3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.

1. \(\quad\) Find \(T\).
2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}

4 A particle moves with constant acceleration between the points \(A\) and \(B\). At \(A\), it has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(B\), it has velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It takes 10 seconds to move from \(A\) to \(B\).

1. Find the acceleration of the particle.
2. Find the distance between \(A\) and \(B\).
3. Find the average velocity as the particle moves from \(A\) to \(B\).

6 Emma is in a park with her dog, Roxy. Emma throws a ball and Roxy catches it in her mouth. The ground in the park is horizontal. Emma throws the ball from a point at a height of 1.2 metres above the ground and Roxy catches the ball when it is at a height of 0.5 metres above the ground. Emma throws the ball with an initial velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal.

1. Find the time that the ball takes to travel from Emma's hand to Roxy's mouth.
2. Find the horizontal distance travelled by the ball during its flight.
3. During the flight, the speed of the ball is a maximum when it is at a height of \(h\) metres above the ground. Write down the value of \(h\).
4. Find the maximum speed of the ball during its flight.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-14_1566_1707_1137_153}

8 Two trains, \(A\) and \(B\), are moving on straight horizontal tracks which run alongside each other and are parallel. The trains both move with constant acceleration. At time \(t = 0\), the fronts of the trains pass a signal. The velocities of the trains are shown in the graph below. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-18_633_1077_475_424}

1. Find the distance between the fronts of the two trains when they have the same velocity and state which train has travelled further from the signal.
2. Find the time when \(A\) has travelled 9 metres further than \(B\).
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-20_2288_1707_221_153}

3 A toy car is placed at the top of a ramp. After the car has been released from rest, it travels a distance of 1.08 metres down the ramp, in a time of 1.2 seconds. Assume that there is no resistance to the motion of the car.

1. Find the magnitude of the acceleration of the car while it is moving down the ramp.
2. Find the speed of the car, when it has travelled 1.08 metres down the ramp.
3. Find the angle between the ramp and the horizontal, giving your answer to the nearest degree.
   [0pt] [4 marks]

5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.

1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of the 7 kg particle just before it hits the ground.
3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
   [0pt] [4 marks]

7 At a school fair, there is a competition in which people try to kick a football so that it lands in a large rectangular box. The height of the top of the box is 1 metre and its width is 3 metres. One student kicks a football so that it initially moves at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. It hits the top front edge of the box, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-16_465_1342_625_351} Model the football as a particle that is not subject to any resistance forces as it moves.

1. Find the time taken for the football to move from the point where it was kicked to the box.
2. Find the horizontal distance from the point where the football is kicked to the front of the box.
3. If the same student kicks the football at the same angle from the same initial position, what is the speed at which the student should kick the football if it is to hit the top back edge of the box?
4. Explain the significance of modelling the football as a particle in this context.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}]{5dd17095-18a6-470b-a24a-4456c8e3ed31-23_2488_1709_219_153}
   \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

2. A particle passes through a point \(O\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) and moves in a straight line with constant acceleration \(3.6 \mathrm {~ms} ^ { - 2 }\) for \(t\) seconds until it reaches the point \(P\). The acceleration is then reduced to \(2 \mathrm {~ms} ^ { - 2 }\) and this is maintained for another \(t\) seconds until the particle passes the point \(Q\) with speed \(16 \mathrm {~ms} ^ { - 1 }\). Calculate

1. the time taken by the particle to travel from \(O\) to \(Q\),
2. the distance \(O Q\).

5. A small stone is projected with speed \(7 \mathrm {~ms} ^ { - 1 }\) from \(P\), the bottom of a rough plane inclined at \(25 ^ { \circ }\) to the horizontal, and moves up a line of greatest slope of the plane until it comes to instantaneous rest at \(Q\), where \(P Q = 4 \mathrm {~m}\).

1. Show that the deceleration of the stone as it moves up the plane has magnitude \(\frac { 49 } { 8 } \mathrm {~ms} ^ { - 2 }\).
2. Find the coefficient of friction between the stone and the plane,
3. Find the speed with which the stone returns to \(P\).
4. Name one force which you have ignored in your mathematical model, and state whether the answer to part (c) would be larger or smaller if that force were taken into account.

7.
A particle \(P\), of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle \(Q\) of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially \(Q\) is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.

1. Find the acceleration of the two particles.
2. Find the speed with which \(Q\) hits the ground. Assuming that \(Q\) does not rebound from the ground while the string is slack,
3. show that \(P\) does not reach the pulley before \(Q\) starts to move again.
4. Find the speed with which \(Q\) leaves the ground when the string again becomes taut.
   (3 marks)

6. A small ring, of mass \(m \mathrm {~kg}\), can slide along a straight wire which is fixed at an angle of \(45 ^ { \circ }\) to the horizontal as shown. The coefficient of friction between the ring and the wire is \(\frac { 2 } { 7 }\).
The ring rests in equilibrium on the wire and is just prevented from \includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-2_273_296_1192_1617}
sliding down the wire when a horizontal string is attached to it, as shown

1. Show that the tension in the string has magnitude \(\frac { 5 m g } { 9 } \mathrm {~N}\). The string is now removed and the ring starts to slide down the wire.
2. Find the time that elapses before the ring has moved 10 cm along the wire.

7. Two cyclists, Alice and Bobbie, travel from \(P\) to \(Q\) along a straight path. Alice starts from rest at \(P\) just as Bobbie passes her at \(3.5 \mathrm {~ms} ^ { - 1 }\). Bobbie continues at this speed while Alice accelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds until she attains her maximum speed. At this moment both cyclists immediately start to slow down, with constant but different decelerations, and they come to rest at \(Q 80\) seconds after Alice started moving.

1. Sketch, on the same diagram, the velocity-time graphs for the two cyclists. By using the fact that both cyclists cover the same distance, find
2. the value of \(T\),
3. the distance between \(P\) and \(Q\),
4. the magnitude of Bobbie's deceleration.