3.02d Constant acceleration: SUVAT formulae

4 A ball \(A\) is released from rest at the top of a tall tower. One second later, another ball \(B\) is projected vertically upwards from ground level near the bottom of the tower with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The two balls are at the same height 1.5 s after ball \(B\) is projected.

1. Show that the height of the tower is 50 m .
2. Find the length of time for which ball \(B\) has been in motion when ball \(A\) reaches the ground. Hence find the total distance travelled by ball \(B\) up to the instant when ball \(A\) reaches the ground.

1 A particle of mass 0.2 kg moving in a straight line experiences a constant resistance force of 1.5 N . When the particle is moving at speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), a constant force of magnitude \(F \mathrm {~N}\) is applied to it in the direction in which it is moving. Given that the speed of the particle 5 seconds later is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(F\).

6 A particle is projected from a point \(P\) with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope \(P Q R\) of a rough inclined plane. The distances \(P Q\) and \(Q R\) are both equal to 0.8 m . The particle takes 0.6 s to travel from \(P\) to \(Q\) and 1 s to travel from \(Q\) to \(R\).

1. Show that the deceleration of the particle is \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and hence find \(u\), giving your answer as an exact fraction.
2. Given that the plane is inclined at \(3 ^ { \circ }\) to the horizontal, find the value of the coefficient of friction between the particle and the plane.

4 A runner sets off from a point \(P\) at time \(t = 0\), where \(t\) is in seconds. The runner starts from rest and accelerates at \(1.2 \mathrm {~ms} ^ { - 2 }\) for 5 s . For the next 12 s the runner moves at constant speed before decelerating uniformly over a period of 3 s , coming to rest at \(Q\). A cyclist sets off from \(P\) at time \(t = 10\) and accelerates uniformly for 10 s , before immediately decelerating uniformly to rest at \(Q\) at time \(t = 30\).

1. Sketch the velocity-time graph for the runner and show that the distance \(P Q\) is 96 m . \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-06_821_1451_708_388}
2. Find the magnitude of the acceleration of the cyclist.

5 \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .

1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\). \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.

2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).

4 A car travels along a straight road with constant acceleration. It passes through points \(P , Q , R\) and \(S\). The times taken for the car to travel from \(P\) to \(Q , Q\) to \(R\) and \(R\) to \(S\) are each equal to 10 s . The distance \(Q R\) is 1.5 times the distance \(P Q\). At point \(Q\) the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the acceleration of the car is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the distance \(Q S\) and hence find the average speed of the car between \(Q\) and \(S\).

5 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg . His power output is a constant 240 W . His acceleration when he is travelling at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that the resistance to the cyclist's motion is 16 N .
2. Find the steady speed that the cyclist can maintain if his power output and the resistance force are both unchanged.
3. The cyclist later ascends a straight hill inclined at \(3 ^ { \circ }\) to the horizontal. His power output and the resistance force are still both unchanged. Find his acceleration when he is travelling at \(4 \mathrm {~ms} ^ { - 1 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.

1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
   At the instant when \(B\) reaches the ground, the string breaks.
2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle is projected with speed \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. The angle of projection is \(\alpha ^ { \circ }\) above the horizontal. The particle reaches the ground again after 10 s .

1. Find the value of \(\alpha\).
2. Find the greatest height reached by the particle.
3. At time \(T\) s after the instant of projection the direction of motion of the particle is at an angle of \(45 ^ { \circ }\) above the horizontal. Find the value of \(T\).

6 A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction upwards at \(35 ^ { \circ }\) to the horizontal. The particle passes through the point \(M\) at time \(T\) seconds after the instant of projection. The point \(M\) is 2 m above the ground and at a horizontal distance of 25 m from \(O\).

1. Find the values of \(V\) and \(T\).
2. Find the speed of the particle as it passes through \(M\) and determine whether it is moving upwards or downwards.

7 A stone is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The stone is at its highest point when it has travelled a horizontal distance of 19.2 m .

1. Find the value of \(V\). After passing through its highest point the stone strikes a vertical wall at a point 4 m above the ground.
2. Find the horizontal distance between \(O\) and the wall. At the instant when the stone hits the wall the horizontal component of the stone's velocity is halved in magnitude and reversed in direction. The vertical component of the stone's velocity does not change as a result of the stone hitting the wall.
3. Find the distance from the wall of the point where the stone reaches the ground.

7 A particle is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground, in a direction making an angle of \(\alpha ^ { \circ }\) above the horizontal. The particle reaches the ground again after 12 s . Find

1. the value of \(\alpha\),
2. the greatest height reached by the particle,
3. the length of time for which the direction of motion of the particle is between \(20 ^ { \circ }\) above the horizontal and \(20 ^ { \circ }\) below the horizontal,
4. the horizontal distance travelled by the particle in the time found in part (iii).

5 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-4_547_933_269_607} Particles \(A\) and \(B\) are projected simultaneously from the top \(T\) of a vertical tower, and move in the same vertical plane. \(T\) is 7.2 m above horizontal ground. \(A\) is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is projected at an angle of \(60 ^ { \circ }\) above the horizontal with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } . A\) and \(B\) move away from each other (see diagram).

1. Find the time taken for \(A\) to reach the ground. At the instant when \(A\) hits the ground,
2. show that \(B\) is approximately 5.2 m above the ground,
3. find the distance \(A B\).

5 A small stone is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. Referred to horizontal and vertically upwards axes through \(O\), the equation of the stone's trajectory is \(y = 0.75 x - 0.02 x ^ { 2 }\), where \(x\) and \(y\) are in metres. Find

1. the values of \(\theta\) and \(V\),
2. the distance from \(O\) of the point where the stone hits the ground,
3. the greatest height reached by the stone.

5 A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).

1. Using the equation of the particle's trajectory and the identity \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\), show that the possible values of \(\tan \theta\) are \(\frac { 3 } { 4 }\) and \(\frac { 17 } { 4 }\).
2. Find the distance \(O A\) for each of the two possible values of \(\tan \theta\).
3. Sketch in the same diagram the two possible trajectories.

1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.

1. Calculate the height of the top of the post above the ground.
2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.

3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = 2 x - x ^ { 2 }$$

1. Find the value of \(x\) for the point where \(P\) strikes the plane.
2. Find the angle and speed of projection of \(P\).
3. Calculate the speed of \(P\) immediately before it strikes the plane.

4 A small ball is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
2. Find \(x\) for the position of the ball when its path makes an angle of \(15 ^ { \circ }\) below the horizontal. [4]

1 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the vertically upwards displacement of the ball from \(O\) is \(\left( 14 t - k t ^ { 2 } \right) \mathrm { m }\), where \(k\) is a constant.

1. State the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-03_56_1563_495_331}
2. Show that the initial speed of the ball is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the horizontal displacement of the ball from \(O\) when \(t = 3\).

6 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(\theta\).
2. Show that at the instant 4 s after projection the particle is 33.75 m below the level of the point of projection and find the direction of motion at this instant.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-12_259_609_255_769} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows an object made from a uniform wire of length 0.8 m . The object consists of a straight part \(A B\), and a semicircular part \(B C\) such that \(A , B\) and \(C\) lie in the same straight line. The radius of the semicircle is \(r \mathrm {~m}\) and the centre of mass of the object is 0.1 m from line \(A B C\).
3. Show that \(r = 0.2\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-13_615_383_260_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The object is freely suspended at \(A\) and a horizontal force of magnitude 7 N is applied to the object at \(C\) so that the object is in equilibrium with \(A B C\) vertical (see Fig. 2).
4. Calculate the weight of the object.
   The 7 N force is removed and the object hangs in equilibrium with \(A B C\) at an angle of \(\theta ^ { \circ }\) with the vertical.
5. Find \(\theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 A particle is projected from a point on horizontal ground. At the instant 2 s after projection, the particle has travelled a horizontal distance of 30 m and is at its greatest height above the ground. Find the initial speed and the angle of projection of the particle.

1 A particle is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle at the instant 4 s after projection.

3 A small ball is projected from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively, where \(x = 4 t\) and \(y = 6 t - 5 t ^ { 2 }\).

1. Find the equation of the trajectory of the ball.
2. Hence or otherwise calculate the angle of projection of the ball and its initial speed.

1 A stone is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The horizontal and vertically upward displacements of the stone from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Find the equation of the stone's trajectory. The stone enters the sea at a horizontal distance of 24 m from the base of the cliff.
2. Find the height above sea level of the top of the cliff.