3.02d Constant acceleration: SUVAT formulae

7 A particle \(P _ { 1 }\) is projected vertically upwards, from horizontal ground, with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle \(P _ { 2 }\) is projected vertically upwards from the top of a tower of height 25 m , with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the time for which \(P _ { 1 }\) is higher than the top of the tower,
2. the velocities of the particles at the instant when the particles are at the same height,
3. the time for which \(P _ { 1 }\) is higher than \(P _ { 2 }\) and is moving upwards.

1 A car of mass 1200 kg travels on a horizontal straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that the car's speed increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 525 m , find the value of \(a\). The car's engine exerts a constant driving force of 900 N . The resistance to motion of the car is constant and equal to \(R \mathrm {~N}\).
2. Find \(R\).

4 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_568_1084_269_532} The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point \(A\) above the surface of liquid in a container. The downward velocity of the stone \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone hits the surface of the liquid with velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0.7\). It reaches the bottom of the container with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 1.2\).

1. Find
   1. the height of \(A\) above the surface of the liquid,
   2. the depth of liquid in the container.
   3. Find the deceleration of the stone while it is moving in the liquid.
   4. Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N , find the mass of the stone.

7 Two particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. The particles start at the same instant and from the same point, each with speed \(1.3 \mathrm {~ms} ^ { - 1 }\). Initially \(P\) moves down the plane and \(Q\) moves up the plane. The distance between the particles \(t\) seconds after they start to move is \(d \mathrm {~m}\).

1. Show that \(d = 2.6 t\). When \(t = 2.5\) the difference in the vertical height of the particles is 1.6 m . Find
2. the acceleration of the particles down the plane,
3. the distance travelled by \(P\) when \(Q\) is at its highest point.

1 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_203_1200_264_475} A particle slides up a line of greatest slope of a smooth plane inclined at an angle \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the points \(A\) and \(B\) with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 4 m (see diagram). Find

1. the deceleration of the particle,
2. the value of \(\alpha\).

4 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_702_709_269_719} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed peg. The particles are held at rest with the string taut. Both particles are at a height of 0.9 m above the ground (see diagram). The system is released and each of the particles moves vertically. Find

1. the acceleration of \(P\) and the tension in the string before \(P\) reaches the ground,
2. the time taken for \(P\) to reach the ground.

6 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_593_746_269_701} A particle \(P\) starts from rest at the point \(A\) and travels in a straight line, coming to rest again after 10 s . The velocity-time graph for \(P\) consists of two straight line segments (see diagram). A particle \(Q\) starts from rest at \(A\) at the same instant as \(P\) and travels along the same straight line as \(P\). The velocity of \(Q\) is given by \(v = 3 t - 0.3 t ^ { 2 }\) for \(0 \leqslant t \leqslant 10\). The displacements from \(A\) of \(P\) and \(Q\) are the same when \(t = 10\).

1. Show that the greatest velocity of \(P\) during its motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(t\), in the interval \(0 < t < 5\), for which the acceleration of \(Q\) is the same as the acceleration of \(P\).

1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the particle,
2. the value of \(\alpha\).

5 A ball moves on the horizontal surface of a billiards table with deceleration of constant magnitude \(d \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball starts at \(A\) with speed \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaches the edge of the table at \(B , 1.2 \mathrm {~s}\) later, with speed \(1.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the distance \(A B\) and the value of \(d\). \(A B\) is at right angles to the edge of the table containing \(B\). The table has a low wall along each of its edges and the ball rebounds from the wall at \(B\) and moves directly towards \(A\). The ball comes to rest at \(C\) where the distance \(B C\) is 2 m .
2. Find the speed with which the ball starts to move towards \(A\) and the time taken for the ball to travel from \(B\) to \(C\).
3. Sketch a velocity-time graph for the motion of the ball, from the time the ball leaves \(A\) until it comes to rest at \(C\), showing on the axes the values of the velocity and the time when the ball is at \(A\), at \(B\) and at \(C\).

6 Particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. \(P\) is released from rest at a point \(O\) on the line and 2 s later passes through the point \(A\) with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) and the angle of inclination of the plane. At the instant that \(P\) passes through \(A\) the particle \(Q\) is released from rest at \(O\). At time \(t\) s after \(Q\) is released from \(O\), the particles \(P\) and \(Q\) are 4.9 m apart.
2. Find the value of \(t\).

5 A train starts from rest at a station \(A\) and travels in a straight line to station \(B\), where it comes to rest. The train moves with constant acceleration \(0.025 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 600 s , with constant speed for the next 2600 s , and finally with constant deceleration \(0.0375 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the total time taken for the train to travel from \(A\) to \(B\).
2. Sketch the velocity-time graph for the journey and find the distance \(A B\).
3. The speed of the train \(t\) seconds after leaving \(A\) is \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). State the possible values of \(t\).

7 Loads \(A\) and \(B\), of masses 1.2 kg and 2.0 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical. \(A\) is released and starts to move upwards. It does not reach the pulley in the subsequent motion.

1. Find the acceleration of \(A\) and the tension in the string.
2. Find, for the first 1.5 metres of \(A\) 's motion,
   1. A's gain in potential energy,
   2. the work done on \(A\) by the tension in the string,
   3. A's gain in kinetic energy. B hits the floor 1.6 seconds after \(A\) is released. \(B\) comes to rest without rebounding and the string becomes slack.
   4. Find the time from the instant the string becomes slack until it becomes taut again.

3 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-2_748_1410_979_370} The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:

• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.
  1. Show that the total distance fallen is 1048 m .

The weight of the parachutist is 850 N .

Find the upward force on the parachutist due to the parachute, during the second stage.

5 Two particles \(P\) and \(Q\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(P\) and \(Q\) are \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively and the heights of \(P\) and \(Q\) above the ground, \(t\) seconds after projection, are \(h _ { P } \mathrm {~m}\) and \(h _ { Q } \mathrm {~m}\) respectively. Each particle comes to rest on returning to the ground.

1. Find the set of values of \(t\) for which the particles are travelling in opposite directions.
2. At a certain instant, \(P\) and \(Q\) are above the ground and \(3 h _ { P } = 8 h _ { Q }\). Find the velocities of \(P\) and \(Q\) at this instant.

7 A walker travels along a straight road passing through the points \(A\) and \(B\) on the road with speeds \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The walker's acceleration between \(A\) and \(B\) is constant and equal to \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time taken by the walker to travel from \(A\) to \(B\), and find the distance \(A B\). A cyclist leaves \(A\) at the same instant as the walker. She starts from rest and travels along the straight road, passing through \(B\) at the same instant as the walker. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant.
2. Show that when \(t = 64.05\) the speed of the walker and the speed of the cyclist are the same, correct to 3 significant figures.
3. Find the cyclist's acceleration at the instant she passes through \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-2_443_825_755_661} A particle \(P\) is projected from the top of a smooth ramp with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and travels down a line of greatest slope. The ramp has length 6.4 m and is inclined at \(30 ^ { \circ }\) to the horizontal. Another particle \(Q\) is released from rest at a point 3.2 m vertically above the bottom of the ramp, at the same instant that \(P\) is projected (see diagram). Given that \(P\) and \(Q\) reach the bottom of the ramp simultaneously, find

1. the value of \(u\),
2. the speed with which \(P\) reaches the bottom of the ramp. \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-3_609_1539_255_303} The diagram shows the velocity-time graphs for the motion of two particles \(P\) and \(Q\), which travel in the same direction along a straight line. \(P\) and \(Q\) both start at the same point \(X\) on the line, but \(Q\) starts to move \(T\) s later than \(P\). Each particle moves with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first 20 s of its motion. The speed of each particle changes instantaneously to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after it has been moving for 20 s and the particle continues at this speed.

6 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_465_849_1475_648} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle \(\theta\) with the ground, where \(\sin \theta = 0.8\). Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.

1. Find the tension in the string and the acceleration of the particles while both are moving. The speed of \(P\) when it reaches the ground is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On reaching the ground \(P\) comes to rest and remains at rest. \(Q\) continues to move up the slope but does not reach the pulley.
2. Find the time taken from the instant that the particles are released until \(Q\) reaches its greatest height above the ground.

7 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find

1. the acceleration of \(B\) while it is moving downwards,
2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground. \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.

3 The top of a cliff is 40 metres above the level of the sea. A man in a boat, close to the bottom of the cliff, is in difficulty and fires a distress signal vertically upwards from sea level. Find

1. the speed of projection of the signal given that it reaches a height of 5 m above the top of the cliff,
2. the length of time for which the signal is above the level of the top of the cliff. The man fires another distress signal vertically upwards from sea level. This signal is above the level of the top of the cliff for \(\sqrt { } ( 17 ) \mathrm { s }\).
3. Find the speed of projection of the second signal.

5 A particle \(P\) is projected vertically upwards from a point on the ground with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(Q\) is projected vertically upwards from the same point with speed \(7 \mathrm {~ms} ^ { - 1 }\). Particle \(Q\) is projected \(T\) seconds later than particle \(P\).

1. Given that the particles reach the ground at the same instant, find the value of \(T\).
2. At a certain instant when both \(P\) and \(Q\) are in motion, \(P\) is 5 m higher than \(Q\). Find the magnitude and direction of the velocity of each of the particles at this instant.

6 A particle \(P\) of mass 0.2 kg is released from rest at a point 7.2 m above the surface of the liquid in a container. \(P\) falls through the air and into the liquid. There is no air resistance and there is no instantaneous change of speed as \(P\) enters the liquid. When \(P\) is at a distance of 0.8 m below the surface of the liquid, \(P\) 's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only force on \(P\) due to the liquid is a constant resistance to motion of magnitude \(R \mathrm {~N}\).

1. Find the deceleration of \(P\) while it is falling through the liquid, and hence find the value of \(R\). The depth of the liquid in the container is \(3.6 \mathrm {~m} . P\) is taken from the container and attached to one end of a light inextensible string. \(P\) is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of \(R \mathrm {~N}\) continues to act. The particle reaches the surface 4 s after leaving the bottom of the container.
2. Find the tension in the string.

2 A car of mass 1250 kg travels up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.02\). The power provided by the car's engine is 23 kW . The resistance to motion is constant and equal to 600 N . Find the speed of the car at an instant when its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1 One end of a light inextensible string is attached to a block. The string makes an angle of \(60 ^ { \circ }\) above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the string is 8 N . The block starts to move with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 5 s of the block's motion, find

1. the distance travelled,
2. the work done by the tension in the string.

5 A particle \(P\) starts from rest at a point \(O\) on a horizontal straight line. \(P\) moves along the line with constant acceleration and reaches a point \(A\) on the line with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the instant that \(P\) leaves \(O\), a particle \(Q\) is projected vertically upwards from the point \(A\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(P\) and \(Q\) collide at \(A\). Find

1. the acceleration of \(P\),
2. the distance \(O A\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_538_414_315_370} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-3_561_686_264_1080} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Two particles \(P\) and \(Q\) have masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively. The particles are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. \(P\) is held at rest with the string taut and both straight parts of the string vertical. \(P\) and \(Q\) are each at a height of \(h \mathrm {~m}\) above horizontal ground (see Fig. 1). \(P\) is released and \(Q\) moves downwards. Subsequently \(Q\) hits the ground and comes to rest. Fig. 2 shows the velocity-time graph for \(P\) while \(Q\) is moving downwards or is at rest on the ground.

1 A lift moves upwards from rest and accelerates at \(0.9 \mathrm {~ms} ^ { - 2 }\) for 3 s . The lift then travels for 6 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 4 s .

1. Sketch a velocity-time graph for the motion.
2. Find the total distance travelled by the lift.