3.02d Constant acceleration: SUVAT formulae

6. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 0.5 kg and \(m \mathrm {~kg}\) respectively, where \(m < 0.5\). The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially \(P\) is 3.15 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. After \(P\) has been descending for 1.5 s , it strikes the ground. Particle \(P\) reaches the ground before \(Q\) has reached the pulley.

1. Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string as \(P\) descends.
3. Show that \(m = \frac { 5 } { 18 }\).
4. State how you have used the information that the string is inextensible. When \(P\) strikes the ground, \(P\) does not rebound and the string becomes slack. Particle \(Q\) then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
5. Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.

2. At time \(t = 0\), a particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above the ground. At time \(T\) seconds, the particle hits the ground with speed \(17.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(u\),
2. the value of \(T\).

4. A car is moving along a straight horizontal road. The speed of the car as it passes the point \(A\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car maintains this speed for 30 s . The car then decelerates uniformly to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is then maintained until the car passes the point \(B\). The time taken to travel from \(A\) to \(B\) is 90 s and \(A B = 1410 \mathrm {~m}\).

1. Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
2. Calculate the deceleration of the car as it decelerates from \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Question 4 continued \(\_\_\_\_\)

3. A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 20 seconds. It moves at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 30 seconds, then moves with constant deceleration \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves at speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 15 seconds and then moves with constant deceleration \(\frac { 1 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.

1. Sketch, in the space below, a speed-time graph for this journey. In the first 20 seconds of this journey the car travels 140 m . Find
2. the value of \(V\),
3. the total time for this journey,
4. the total distance travelled by the car.

5. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.

1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
3. find the magnitude of the impulse on \(B\) due to the impact with the plane.

2. \begin{figure}[h] \end{figure} A rough plane is inclined at \(40 ^ { \circ }\) to the horizontal. Two points \(A\) and \(B\) are 3 metres apart and lie on a line of greatest slope of the inclined plane, with \(A\) above \(B\), as shown in Figure 2. A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the plane at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is released.

1. Find the acceleration of \(P\) down the plane.
2. Find the speed of \(P\) at \(B\).

7. \begin{figure}[h] \end{figure} Three particles \(A , B\) and \(C\) have masses \(3 m , 2 m\) and \(2 m\) respectively. Particle \(C\) is attached to particle \(B\). Particles \(A\) and \(B\) are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 5. The system is released from rest and \(A\) moves upwards.

1. 1. Show that the acceleration of \(A\) is \(\frac { g } { 7 }\)
   2. Find the tension in the string as \(A\) ascends. At the instant when \(A\) is 0.7 m above its original position, \(C\) separates from \(B\) and falls away. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(A\) at the instant when it is 0.7 m above its original position.
3. Find the acceleration of \(A\) at the instant after \(C\) separates from \(B\).
4. Find the greatest height reached by \(A\) above its original position. \includegraphics[max width=\textwidth, alt={}, center]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-14_115_161_2455_1784}

2. A small stone is projected vertically upwards from a point \(O\) with a speed of \(19.6 \mathrm {~ms} ^ { - 1 }\). Modelling the stone as a particle moving freely under gravity,

1. find the greatest height above \(O\) reached by the stone,
2. find the length of time for which the stone is more than 14.7 m above \(O\).

7. A train travels along a straight horizontal track between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~ms} ^ { - 1 } , ( V < 50 )\). The train then travels at this constant speed before it moves with constant deceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(B\).

1. Sketch in the space below a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). The total time for the journey from \(A\) to \(B\) is 5 minutes.
2. Find, in terms of \(V\), the length of time, in seconds, for which the train is
   1. accelerating,
   2. decelerating,
   3. moving with constant speed. Given that the distance between the two stations \(A\) and \(B\) is 6.3 km ,
3. find the value of \(V\).

4. Two trains \(M\) and \(N\) are moving in the same direction along parallel straight horizontal tracks. At time \(t = 0 , M\) overtakes \(N\) whilst they are travelling with speeds \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Train \(M\) overtakes train \(N\) as they pass a point \(X\) at the side of the tracks. After overtaking \(N\), train \(M\) maintains its speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds and then decelerates uniformly, coming to rest next to a point \(Y\) at the side of the tracks. After being overtaken, train \(N\) maintains its speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s and then decelerates uniformly, also coming to rest next to the point \(Y\). The times taken by the trains to travel between \(X\) and \(Y\) are the same.

1. Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between \(X\) and \(Y\). Given that \(X Y = 975 \mathrm {~m}\),
2. find the value of \(T\).
   

6. A cyclist is moving along a straight horizontal road and passes a point \(A\). Five seconds later, at the instant when she is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), she passes the point \(B\). She moves with constant acceleration from \(A\) to \(B\). Given that \(A B = 40 \mathrm {~m}\), find

1. the acceleration of the cyclist as she moves from \(A\) to \(B\),
2. the time it takes her to travel from \(A\) to the midpoint of \(A B\).

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
4. find \(d\) in terms of \(h\).
5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)

   (Total 15 marks)	Leave blank Q8

4. A ball of mass 0.2 kg is projected vertically downwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

1. the value of \(U\). After hitting the ground, the ball moves vertically upwards and passes through a point \(B\) which is 1 m above the ground.
2. Find the time between the instant when the ball hits the ground and the instant when the ball first passes through \(B\).
3. Sketch a velocity-time graph for the motion of the ball from when it was projected from \(A\) to when it first passes through \(B\). (You need not make any further calculations to draw this sketch.)

3. A car accelerates uniformly from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in \(T\) seconds. The car then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(4 T\) seconds and finally decelerates uniformly to rest in a further 50 s .

1. Sketch a speed-time graph to show the motion of the car. The total distance travelled by the car is 1220 m . Find
2. the value of \(T\),
3. the initial acceleration of the car.

5. A racing car is moving along a straight horizontal track with constant acceleration. There are three checkpoints, \(P , Q\) and \(R\), on the track, where \(P Q = 48 \mathrm {~m}\) and \(Q R = 200 \mathrm {~m}\). The car takes 3 s to travel from \(P\) to \(Q\) and 5 s to travel from \(Q\) to \(R\). Find

1. the acceleration of the car,
2. the speed of the car as it passes \(P\).

8. Two trains, \(A\) and \(B\), start together from rest, at time \(t = 0\), at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train \(A\) moves with constant acceleration \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train \(B\) moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .

1. Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
2. Find the acceleration of train \(B\) for the first half of its journey.
3. Find the times when the two trains are moving at the same speed.
4. Find the distance between the trains 96 s after they start. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}

1. A train moves along a straight horizontal track between two stations \(R\) and \(S\). Initially the train is at rest at \(R\). The train accelerates uniformly at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest at \(R\) until it is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the next 200 seconds the train maintains a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train then decelerates uniformly at \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(S\).

Find

1. the time taken by the train to travel from \(R\) to \(S\),
2. the distance from \(R\) to \(S\),
3. the average speed of the train during the journey from \(R\) to \(S\).

5. A cyclist is travelling along a straight horizontal road. The cyclist starts from rest at point \(A\) on the road and accelerates uniformly at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 20 seconds. He then moves at constant speed for \(4 T\) seconds, where \(T < 20\). He then decelerates uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and after \(T\) seconds passes through point \(B\) on the road. The distance from \(A\) to \(B\) is 705 m .

1. Sketch a speed-time graph for the motion of the cyclist between points \(A\) and \(B\).
2. Find the value of \(T\). The cyclist continues his journey, still decelerating uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), until he comes to rest at point \(C\) on the road.
3. Find the total time taken by the cyclist to travel from \(A\) to \(C\).

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(A\) of mass 3 kg . Block \(A\) is held at rest on a smooth fixed plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal ground. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a block \(B\) of mass 5 kg . Block \(B\) hangs freely at rest below the pulley, as shown in Figure 4. The system is released from rest with the string taut. By modelling the two blocks as particles,

1. find the tension in the string as \(B\) descends. After falling for 1.5 s , block \(B\) hits the ground and is immediately brought to rest. In its subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(B\) at the instant it hits the ground.
3. Find the total distance moved up the plane by \(A\) before it comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_97_141_2519_1804} \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_125_161_2624_1779}

6. A train travels for a total of 270 s along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration for 60 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train then travels at this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it moves with constant deceleration for 30 s , coming to rest at \(B\).

1. Sketch below a speed-time graph for the journey of the train between the two stations \(A\) and \(B\). Given that the distance between the two stations is 4.5 km ,
2. find the value of \(V\),
3. find how long it takes the train to travel from station \(A\) to the point that is exactly halfway between the two stations. The train is travelling at speed \(\frac { 1 } { 4 } V \mathrm {~ms} ^ { - 1 }\) at times \(T _ { 1 }\) seconds and \(T _ { 2 }\) seconds after leaving station \(A\).
4. Find the value of \(T _ { 1 }\) and the value of \(T _ { 2 }\)