3.02d Constant acceleration: SUVAT formulae

\includegraphics{figure_2} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a\) m s\(^{-2}\) for 3.5 s, reaching a speed of 14 m s\(^{-1}\). Car \(A\) then moves with constant speed 14 m s\(^{-1}\).

1. Find the value of \(a\). [2] Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with constant acceleration of 3 m s\(^{-2}\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]
3. Find the value of \(T\). [8]
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\). [1]
5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). [3]

A small ball is projected vertically upwards from a point \(O\) with speed 14.7 m s\(^{-1}\). The point \(O\) is 2.5 m above the ground. The motion of the ball is modelled as that of a particle moving freely under gravity. Find

1. the maximum height above the ground reached by the ball, [4]
2. the time taken for the ball to first reach a height of 1 m above the ground, [4]
3. the speed of the ball at the instant before it strikes the ground for the first time. [3]

An athlete goes for a run along a straight horizontal road. Starting from rest, she accelerates at 0.6 m s\(^{-2}\) up to a speed of \(V\) m s\(^{-1}\). She then maintains this constant speed of \(V\) m s\(^{-1}\) before finally decelerating at 0.2 m s\(^{-2}\) back to rest. She covers a total distance of 1500 m in 270 s.

1. Sketch a speed-time graph to represent the athlete's run. [2]
2. Show that she accelerates for \(\frac{5V}{3}\) seconds. [2]
3. Show that \(V^2 - kV + 450 = 0\), where \(k\) is a constant to be found. [6]
4. Find the value of \(V\), justifying your answer. [4]

\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth light pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,

1. write down an equation of motion for each particle. [4]
2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
3. Explain how you have used the fact that the string is inextensible in your calculation. [1]

When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.

1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]

A small ball is projected vertically upwards with speed \(29.4\text{ ms}^{-1}\) from a point \(A\) which is \(19.6\text{ m}\) above horizontal ground. The ball is modelled as a particle moving freely under gravity until it hits the ground. It is assumed that the ball does not rebound.

1. Find the distance travelled by the ball while its speed is less than \(14.7\text{ ms}^{-1}\) [3]
2. Find the time for which the ball is moving with a speed of more than \(29.4\text{ ms}^{-1}\) [3]
3. Sketch a speed-time graph for the motion of the ball from the instant when it is projected from \(A\) to the instant when it hits the ground. Show clearly where your graph meets the axes. [3]

\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane. The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\) Particle \(Q\) is released from rest. The tension in the string before \(Q\) hits the pulley is \(kmg\), where \(k\) is a constant.

1. Find \(k\) in terms of \(\mu\). [7] Given that \(Q\) is initially a distance \(d\) from the pulley,
2. find, in terms of \(d\), \(g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley. [4]
3. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer. [2]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = 7\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

A railway truck \(A\) of mass 1800 kg is moving along a straight horizontal track with speed 4 m s\(^{-1}\). It collides directly with a stationary truck \(B\) of mass 1200 kg on the same track. In the collision, \(A\) and \(B\) are coupled and move off together.

1. Find the speed of the trucks immediately after the collision. [3]

After the collision, the trucks experience a constant resistive force of magnitude \(R\) newtons. They come to rest 8 s after the collision.

1. Find \(R\). [3]

A racing car moves with constant acceleration along a straight horizontal road. It passes the point \(O\) with speed 12 m s\(^{-1}\). It passes the point \(A\) 4 s later with speed 60 m s\(^{-1}\).

1. Show that the acceleration of the car is 12 m s\(^{-2}\). [2]
2. Find the distance \(OA\). [3]

The point \(B\) is the mid-point of \(OA\).

1. Find, to 3 significant figures, the speed of the car when it passes \(B\). [3]

A motor scooter and a van set off along a straight road. They both start from rest at the same time and level with each other. The scooter accelerates with constant acceleration until it reaches its top speed of 20 m s\(^{-1}\). It then maintains a constant speed of 20 m s\(^{-1}\). The van accelerates with constant acceleration for 10 s until it reaches its top speed \(V\) m s\(^{-1}\), \(V > 20\). It then maintains a constant speed of \(V\) m s\(^{-1}\). The van draws level with the scooter when the scooter has been travelling for 40 s at its top speed. The total distance travelled by each vehicle is then 850 m.

1. Sketch on the same diagram the speed-time graphs of both vehicles to illustrate their motion from the time when they start to the time when the van overtakes the scooter. [3]
2. Find the time for which the scooter is accelerating. [3]
3. Find the top speed of the van. [3]

A train starts from rest at a station \(A\) and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s\(^{-2}\). For the next 24 s it moves at a constant acceleration 0.75 m s\(^{-2}\). It then moves with constant speed for \(T\) seconds. Finally it slows down with constant deceleration 3 m s\(^{-2}\) until it comes to a rest at station \(B\).

1. Show that, 34 s after leaving \(A\), the speed of the train is 30 m s\(^{-1}\). [3]
2. Sketch a speed-time graph to illustrate the motion of the train as it moves from \(A\) to \(B\). [3]
3. Find the distance moved by the train during the first 34 s of its journey from \(A\). [4]

The distance from \(A\) to \(B\) is 3 km.

1. Find the value of \(T\). [4]

\includegraphics{figure_2} A sprinter runs a race of 200 m. Her total time for running the race is 25 s. Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of 9 m s\(^{-1}\) in 4 s. The speed of 9 m s\(^{-1}\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u\) m s\(^{-1}\) at the end of the race. Calculate

1. the distance covered by the sprinter in the first 20 s of the race, [2]
2. the value of \(u\), [4]
3. the deceleration of the sprinter in the last 5 s of the race. [3]

\includegraphics{figure_4} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s. Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\), [3]
2. the tension in the string, [4]
3. the value of \(\mu\). [5]
4. State how in your calculations you have used the information that the string is inextensible. [1]

A stone \(S\) is sliding on ice. The stone is moving along a straight horizontal line \(ABC\), where \(AB = 24\) m and \(AC = 30\) m. The stone is subject to a constant resistance to motion of magnitude 0.3 N. At \(A\) the speed of \(S\) is 20 m s\(^{-1}\), and at \(B\) the speed of \(S\) is 16 m s\(^{-1}\). Calculate

1. the deceleration of \(S\), [2]
2. the speed of \(S\) at \(C\). [3]
3. Show that the mass of \(S\) is 0.1 kg. [2]

At \(C\), the stone \(S\) hits a vertical wall, rebounds from the wall and then slides back along the line \(CA\). The magnitude of the impulse of the wall on \(S\) is 2.4 Ns and the stone continues to move against a constant resistance of 0.3 N.

1. Calculate the time between the instant that \(S\) rebounds from the wall and the instant that \(S\) comes to rest. [6]

A stone is thrown vertically upwards with speed \(16 \text{ m s}^{-1}\) from a point \(h\) metres above the ground. The stone hits the ground \(4\) s later. Find

1. the value of \(h\), [3]
2. the speed of the stone as it hits the ground. [3]

A ball is projected vertically upwards with speed 21 m s\(^{-1}\) from a point \(A\), which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modelling the ball as a particle, find

1. the greatest height above \(A\) reached by the ball, [3]
2. the speed of the ball as it reaches the ground, [3]
3. the time between the instant when the ball is projected from \(A\) and the instant when the ball reaches the ground. [4]

\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.

1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
3. Find the tension in the string. [2]
4. State where in your calculations you have used the information that the string is inextensible. [1]

On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find

1. the speed of \(Q\) as it reaches the ground, [2]
2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]

An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s\(^{-1}\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.

1. In the space below, sketch a speed-time graph to illustrate the motion of the athlete. [3]
2. Calculate the value of \(T\). [5]

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

1. the acceleration of the particle, [3]
2. the coefficient of friction between the particle and the plane. [5]

The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}

1. Find the value of \(X\). [7]

\includegraphics{figure_4} Two particles \(A\) and \(B\) have masses \(5m\) and \(km\) respectively, where \(k < 5\). 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 vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac{1}{4}g\).

1. Show that the tension in the string as \(A\) descends is \(\frac{15}{4}mg\). [3]
2. Find the value of \(k\). [3]
3. State how you have used the information that the pulley is smooth. [1]

After descending for 1.2 s, the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.

1. Find the greatest height reached by \(B\) above the plane. [7]

A lifeboat slides down a straight ramp inclined at an angle of \(15°\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s\(^{-1}\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp. [9]

\includegraphics{figure_4} The velocity-time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart. The train \(P\) leaves the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s\(^{-1}\). The train then maintains this speed for 7 seconds before decelerating uniformly at 1.25 m s\(^{-2}\), coming to rest at the next station.

1. Find the acceleration of \(P\) during the first 300 m of its journey. [2]
2. Find the value of \(T\). [5]

A second train \(Q\) completes the same journey in the same total time. The train leaves the first station, accelerating uniformly from rest until it reaches a speed of \(V\) m s\(^{-1}\) and then immediately decelerates uniformly until it comes to rest at the next station.

1. Sketch on the diagram above, a velocity-time graph which represents the journey of train \(Q\). [2]
2. Find the value of \(V\). [6]

A car moves with constant acceleration along a straight horizontal road. The car passes the point \(A\) with speed \(5 \text{ m s}^{-1}\) and \(4 \text{ s}\) later it passes the point \(B\), where \(AB = 50\text{m}\).

1. Find the acceleration of the car. [3]

When the car passes the point \(C\), it has speed \(30 \text{ m s}^{-1}\).

1. Find the distance \(AC\). [3]

A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of \(27 \text{ m}\). The lift initially accelerates with a constant acceleration of \(2 \text{ m s}^{-1}\) until it reaches a speed of \(3 \text{ m s}^{-1}\). It then moves with a constant speed of \(3 \text{ m s}^{-1}\) for \(T\) seconds. Finally it decelerates with a constant deceleration for \(2.5 \text{ s}\) before coming to rest at the top floor.

1. Sketch a speed-time graph for the motion of the lift. [2]
2. Hence, or otherwise, find the value of \(T\). [3]
3. Sketch an acceleration-time graph for the motion of the lift. [3]

The mass of the man is \(80 \text{ kg}\) and the mass of the lift is \(120 \text{ kg}\). The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find

1. the tension in the cable when the lift is accelerating, [3]
2. the magnitude of the force exerted by the lift on the man during the last \(2.5 \text{ s}\) of the motion. [3]

A particle \(P\) is moving with constant acceleration along a straight horizontal line \(ABC\), where \(AC = 24\) m. Initially \(P\) is at \(A\) and is moving with speed \(5\) m s\(^{-1}\) in the direction \(AB\). After \(1.5\) s, the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5\) m s\(^{-1}\).

1. Show that the speed of \(P\) at \(C\) is \(13\) m s\(^{-1}\). [4]

The mass of \(P\) is \(2\) kg. When \(P\) reaches \(C\), an impulse of magnitude \(30\) Ns is applied to \(P\) in the direction \(CB\).

1. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant. [3]