3.02d Constant acceleration: SUVAT formulae

4 A stone is projected from a point on horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\). At time 1.2 s after projection the stone passes through the point \(A\). Subsequently the stone passes through the point \(B\), which is at the same height above the ground as \(A\). Find the horizontal distance \(A B\).

6 A particle is projected from a point \(O\) at an angle of \(35 ^ { \circ }\) above the horizontal. At time \(T\) s later the particle passes through a point \(A\) whose horizontal and vertically upward displacements from \(O\) are 8 m and 3 m respectively.

1. By using the equation of the particle's trajectory, or otherwise, find (in either order) the speed of projection of the particle from \(O\) and the value of \(T\).
2. Find the angle between the direction of motion of the particle at \(A\) and the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_476_895_269_625} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross-section of a uniform solid. The cross-section has the shape and dimensions shown. The centre of mass \(C\) of the solid lies in the plane of this cross-section. The distance of \(C\) from \(D E\) is \(y \mathrm {~cm}\).
3. Find the value of \(y\). The solid is placed on a rough plane. The coefficient of friction between the solid and the plane is \(\mu\). The plane is tilted so that \(E F\) lies along a line of greatest slope.
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_375_431_1366_897} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The solid is placed so that \(F\) is higher up the plane than \(E\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 1 } { 2 }\). [3]
5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_376_428_2069_900} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The solid is now placed so that \(E\) is higher up the plane than \(F\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Show that \(\mu < \frac { 5 } { 6 }\). [3]

1 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_113_787_264_680} A light elastic spring of natural length 0.25 m and modulus of elasticity 100 N is held horizontally between two parallel plates. The axis of the spring is at right angles to each of the plates. The horizontal force exerted on the spring by each of the plates is 20 N (see diagram). Find the amount by which the spring is compressed and hence write down the distance between the plates.

5 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_405_545_630_799} A uniform lamina \(A O B\) is in the shape of a sector of a circle with centre \(O\) and radius 0.5 m , and has angle \(A O B = \frac { 1 } { 3 } \pi\) radians and weight 3 N . The lamina is freely hinged at \(O\) to a fixed point and is held in equilibrium with \(A O\) vertical by a force of magnitude \(F \mathrm {~N}\) acting at \(B\). The direction of this force is at right angles to \(O B\) (see diagram). Find

1. the value of \(F\),
2. the magnitude of the force acting on the lamina at \(O\).

3 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_408_1164_248_493} A particle \(P\) is released from rest at a point \(A\) which is 7 m above horizontal ground. At the same instant that \(P\) is released a particle \(Q\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(A\) is 24 m . Particle \(Q\) moves in the vertical plane containing \(O\) and \(A\), with initial speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram). Show that the particles collide.

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t \mathrm {~s}\) are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Starting from the equation of the trajectory given in the List of formulae (MF19), show that $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ When \(\theta = \tan ^ { - 1 } 2 , P\) passes through the point with coordinates \(( 10,16 )\).
2. Show that there is no value of \(\theta\) for which \(P\) can pass through the point with coordinates \(( 18,30 )\).

3 A particle \(P\) is projected with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(\sin \theta\).
2. Find the range of the flight.

4 A particle is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. At the instant 3 s after projection the direction of motion of the particle is \(30 ^ { \circ }\) below the horizontal.

1. Find \(V\).
   ..................................................................................................................................
2. Calculate the distance of the particle from \(O\) at the instant 3 s after projection.

2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.

1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-03_67_1571_438_328}
2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.

1 A particle is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is rising.

3 A particle \(P\) is projected with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 4 s after projection the particle passes through the point \(A\), where \(O A = 40 \mathrm {~m}\) and the line \(O A\) makes an angle of \(30 ^ { \circ }\) with the horizontal. Calculate \(V\) and \(\theta\).

7 A particle \(P\) is projected with speed \(\mathrm { Vms } ^ { - 1 }\) at an angle \(75 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane. It then moves freely under gravity.

1. Show that the total time of flight, in seconds, is \(\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }\).
   A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from \(O\). The particle is projected as before but now strikes the barrier, rebounds and returns to \(O\). The coefficient of restitution between the barrier and the particle is \(\frac { 3 } { 5 }\).
2. Explain why the total time of flight is unchanged.
3. Find an expression for \(V\) in terms of \(g\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

2. A ball is thrown vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(P\) at height \(h\) metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is \(6.45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle.

1. Show that \(u = 0.9\)
2. Find the height above \(P\) to which the ball rises before it starts to fall towards the ground again.
3. Find the value of \(h\).

7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.

1. Find the magnitude of the acceleration of \(B\) immediately after release.
2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.

1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.

Modelling the stone as a particle,

1. show that \(u = 17 \frac { 1 } { 2 }\),
2. find the greatest height above \(A\) reached by the stone,
3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.

3. \begin{figure}[h] \end{figure} A car of mass 1200 kg moves along a straight horizontal road. In order to obey a speed restriction, the brakes of the car are applied for 3 s , reducing the car's speed from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brakes are then released and the car continues at a constant speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 s . Figure 2 shows a sketch of the speed-time graph of the car during the 7 s interval. The graph consists of two straight line segments.

1. Find the total distance moved by the car during this 7 s interval.
2. Explain briefly how the speed-time graph shows that, when the brakes are applied, the car experiences a constant retarding force.
3. Find the magnitude of this retarding force.

3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find

1. her speed when she reaches the surface of the pool,
2. the time taken to reach the surface of the pool.
3. State two physical factors which have been ignored in the model.

6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the frictional force acting on \(P\) as it moves up the plane,
2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous rest.

7. Two trains \(A\) and \(B\) run on parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , A\) starts to move. It moves with constant acceleration for 12 s up to a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then moves at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) starts to move in the same direction as \(A\) when \(t = 40\), where \(t\) is measured in seconds. It accelerates with the same initial acceleration as \(A\), up to a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves at a constant speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) overtakes \(A\) after both trains have reached their maximum speed. Train \(B\) overtakes \(A\) when \(t = T\).

1. Sketch, on the same diagram, the speed-time graphs of both trains for \(0 \leq t \leq T\).
2. Find the value of \(T\).

8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find

1. the acceleration of the truck and the ear,
2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}

3. A train moves along a straight track with constant acceleration. Three telegraph poles are set at equal intervals beside the track at points \(A , B\) and \(C\), where \(A B = 50 \mathrm {~m}\) and \(B C = 50 \mathrm {~m}\). The front of the train passes \(A\) with speed \(22.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it passes \(B\). Find

1. the acceleration of the train,
2. the speed of the front of the train when it passes \(C\),
3. the time that elapses from the instant the front of the train passes \(B\) to the instant it passes \(C\).