3.02d Constant acceleration: SUVAT formulae

3. A particle, \(P\), is projected vertically upwards with speed \(U\) from a fixed point \(O\). At the instant when \(P\) reaches its greatest height \(H\) above \(O\), a second particle, \(Q\), is projected with speed \(\frac { 1 } { 2 } U\) vertically upwards from \(O\).

1. Find \(H\) in terms of \(U\) and \(g\).
2. Find, in terms of \(U\) and \(g\), the time between the instant when \(Q\) is projected and the instant when the two particles collide.
3. Find where the two particles collide. DO NOT WRITEIN THIS AREA \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-08_2666_99_107_1957}

5. A car travels at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) in a straight line along a horizontal racetrack. At time \(t = 0\), the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car. The motorcyclist accelerates at \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 15 s and then accelerates at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a further \(T\) seconds until he catches up with the car.

1. Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time \(t = 0\) to the instant when the motorcyclist catches up with the car. At the instant when \(t = t _ { 1 }\) seconds, the car and the motorcyclist are moving at the same speed.
2. Find the value of \(t _ { 1 }\)
3. Show that \(T ^ { 2 } + k T - 300 = 0\), where \(k\) is a constant to be found. DO NOT WRITEIN THIS AREA
   

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1. A small stone is projected vertically upwards with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point \(O\) which is 5 m above horizontal ground. The stone is modelled as a particle moving freely under gravity.

Find

1. the speed of the stone at the instant when it is 2 m above the ground,
2. the total time between the instant when the stone is projected from \(O\) and the instant when it first strikes the ground.

7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of \(3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds ( \(T < 6\) ), until his speed is reduced to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then moves with this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves \(H\) and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.

1. Find the speed of the parachutist at the instant when his parachute opens.
2. Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves \(H\) to the instant when he hits the ground.
3. Find the value of \(T\).
4. Find, to the nearest metre, the height of the point \(H\) above the ground.
   7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.

1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
2. find, in terms of \(h\), the range of possible values of \(d\).
3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.

4. At time \(t = 0\), a small ball is projected vertically upwards from a point \(A\) which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point \(B\), where \(A B = 19.6 \mathrm {~m}\) and first hits the ground at time \(t = T\) seconds. The ball is modelled as a particle moving freely under gravity.

1. Find the value of \(T\).
2. Sketch a speed-time graph for the motion of the ball from \(t = 0\) to \(t = T\) seconds.
   (No further calculations are needed in order to draw this sketch.)

1. A train travels along a straight horizontal track between two stations \(A\) and \(B\).

The train starts from rest at station \(A\) and accelerates uniformly for \(T\) seconds until it reaches a speed of \(20 \mathrm {~ms} ^ { - 1 }\) The train then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 3 minutes before decelerating uniformly until it comes to rest at station \(B\). The magnitude of the acceleration of the train is twice the magnitude of the deceleration.

1. On the axes below, sketch a speed-time graph to illustrate the motion of the train as it moves from station \(A\) to station \(B\). \includegraphics[max width=\textwidth, alt={}, center]{84c0eead-0a87-4d87-b33d-794a94bb466c-02_670_1422_813_312} If you need to redraw your graph, use the axes on page 3 Stations \(A\) and \(B\) are 4.8 km apart.
2. Find the value of \(T\)
3. Find the acceleration of the train during the first \(T\) seconds of its motion. Only use these axes if you need to redraw your graph. \({ } _ { O } ^ { \substack { \text { speed } \\ \left( \mathrm { ms } ^ { - 1 } \right) } }\)

5. \begin{figure}[h] \end{figure} Three points \(P , Q\) and \(R\) are on a horizontal road where \(P Q R\) is a straight line.
The point \(Q\) is between \(P\) and \(R\), with \(P Q = 6 x\) metres and \(Q R = 5 x\) metres, as shown in Figure 2. A vehicle moves along the road from \(P\) to \(Q\) with constant acceleration.
The vehicle is modelled as a particle.
At time \(t = 0\), the vehicle passes \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\) At time \(t = 12 \mathrm {~s}\), the vehicle passes \(Q\) with speed \(2 u \mathrm {~ms} ^ { - 1 }\) Using the model,

1. show that \(x = 3 u\) As the vehicle passes \(Q\), the acceleration of the vehicle changes instantaneously to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The vehicle continues to move with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\) and passes \(R\) with speed \(3 u \mathrm {~ms} ^ { - 1 }\) Using the model,
2. find the value of \(u\),
3. find the distance travelled by the vehicle during the first 14 seconds after passing \(P\)

8. \begin{figure}[h] \end{figure} A parcel of mass 2 kg is pulled up a rough inclined plane by the action of a constant force. The force has magnitude 18 N and acts at an angle of \(40 ^ { \circ }\) to the plane.
The line of action of the force lies in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 5.
The coefficient of friction between the plane and the parcel is 0.3
The parcel is modelled as a particle \(P\)

1. Find the acceleration of \(P\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, where \(A B = 5 \mathrm {~m}\) and \(B\) is above \(A\). Particle \(P\) passes through \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\).
2. Find the speed of \(P\) as it passes through \(B\). The force of 18 N is removed at the instant \(P\) passes through \(B\). As a result, \(P\) comes to rest at the point \(C\).
3. Determine whether \(P\) will remain at rest at \(C\). You must show all stages of your working clearly.

1. A van travels with constant acceleration along a straight horizontal road.

The van passes a point \(A\) with speed \(u \mathrm {~ms} ^ { - 1 }\) and 20 seconds later passes a point \(B\) with speed \(28 \mathrm {~ms} ^ { - 1 }\) The distance \(A B\) is 400 m .

1. Show that \(u = 12\)
2. Find the time taken for the van to travel from \(A\) to the midpoint of \(A B\). The van has mass 1200 kg .
   During its motion the van experiences a constant resistive force of magnitude 260 N
3. Find the magnitude of the driving force exerted by the engine of the van as it travels from \(A\) to \(B\).

1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.

1. Find the value of \(V\).
2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
3. Find the length of time for which the particle is above \(C\).
4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)

8. \begin{figure}[h] \end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\) The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.

1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
   When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

2. A ball is thrown vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is \(h\) metres above the ground. The ball moves freely under gravity until it hits the ground 5 s later.

1. Find the value of \(h\). A second ball is thrown vertically downwards with speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and moves freely under gravity until it hits the ground. The first ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the second ball hits the ground with speed \(\frac { 3 } { 4 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(w\).

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small smooth fixed pulley at the top of the plane. The particle \(Q\) hangs freely below the pulley and 0.6 m above the ground, as shown in Figure 3. The part of the string from \(P\) to the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut. For the motion before \(Q\) hits the ground,

1. 1. show that the acceleration of \(Q\) is \(\frac { 2 g } { 5 }\),
   2. find the tension in the string. On hitting the ground \(Q\) is immediately brought to rest by the impact.
2. Find the speed of \(P\) at the instant when \(Q\) hits the ground. In its subsequent motion \(P\) does not reach the pulley.
3. Find the total distance moved up the plane by \(P\) before it comes to instantaneous rest.
4. Find the length of time between \(Q\) hitting the ground and \(P\) first coming to instantaneous rest.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is pushed by a constant horizontal force of magnitude 30 N up a line of greatest slope of a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing \(P\) and the line of greatest slope of the plane. The particle \(P\) starts from rest. The coefficient of friction between \(P\) and the plane is \(\mu\). After 2 seconds, \(P\) has travelled a distance of 5.5 m up the plane.

1. Find the acceleration of \(P\) up the plane.
2. Find the value of \(\mu\).

1. A small stone is released from rest from a point \(A\) which is at height \(h\) metres above horizontal ground. Exactly one second later another small stone is projected with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from a point \(B\), which is also at height \(h\) metres above the horizontal ground. The motion of each stone is modelled as that of a particle moving freely under gravity. The two stones hit the ground at the same time.

Find the value of \(h\).

5. A car travelling along a straight horizontal road takes 170 s to travel between two sets of traffic lights at \(A\) and \(B\) which are 2125 m apart. The car starts from rest at \(A\) and moves with constant acceleration until it reaches a speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed before moving with constant deceleration, coming to rest at \(B\). The magnitude of the deceleration is twice the magnitude of the acceleration.

1. Sketch, in the space below, a speed-time graph for the motion of the car between \(A\) and \(B\).
2. Find the deceleration of the car.

3. A cyclist starts from rest at the point \(O\) on a straight horizontal road. The cyclist moves along the road with constant acceleration \(2 \mathrm {~ms} ^ { - 2 }\) for 4 seconds and then continues to move along the road at constant speed. At the instant when the cyclist stops accelerating, a motorcyclist starts from rest at the point \(O\) and moves along the road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the same direction as the cyclist. The motorcyclist has been moving for \(T\) seconds when she overtakes the cyclist.

1. Sketch, on the same axes, a speed-time graph for the motion of the cyclist and a speed-time graph for the motion of the motorcyclist, to the time when the motorcyclist overtakes the cyclist.
2. Find, giving your answer to 1 decimal place, the value of \(T\).

4. A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). A particle of mass 2 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is 0.25

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\). The particle now moves down the plane from \(A\).
3. Find the speed of \(P\) as it passes through \(O\).

2. A small ball is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point that is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity. Find

1. the total time from when the ball is thrown to when it first hits the ground,
2. the speed of the ball immediately before it first hits the ground,
3. the total distance travelled by the ball from when it is thrown to when it first hits the ground.
4. Sketch a velocity-time graph for the motion of the ball from when it is thrown to when it first hits the ground. State the coordinates of the start point and the coordinates of the end point of your graph.
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2. A car moves along a straight horizontal road with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a > 0\) The car is modelled as a particle. At time \(t = 0\), the car passes point \(A\) and is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) In the first three seconds after passing \(A\) the car travels 20 m . In the fourth second after passing \(A\) the car travels 10 m . The speed of the car as it passes point \(B\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the time taken for the car to travel from \(A\) to \(B\).
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1. A fixed rough plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)

A particle of mass 6 kg is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(B\).
2. Find the distance \(A B\). The particle now slides down the plane from \(B\). At the instant when the particle passes through the point \(C\) on the plane, the speed of the particle is again \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the distance \(B C\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-23_2647_1835_118_116}