3.02c Interpret kinematic graphs: gradient and area

\includegraphics{figure_2} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for \(8 \text{ s}\) while cutting and then takes \(3 \text{ s}\) to return to its starting position. Find

1. the acceleration of the tool during the first \(2 \text{ s}\) of the motion, [1]
2. the distance the tool moves forward while cutting, [2]
3. the greatest speed of the tool during the return to its starting position. [2]

A vehicle is moving in a straight line. The velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\) after the vehicle starts is given by $$v = A(t - 0.05t^2) \text{ for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t} \text{ for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is \(225 \text{ m}\).

1. Find the value of \(A\) and show that \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of \(315 \text{ m}\). [3]

\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.

1. Find an expression for the maximum height reached by the particle. [3]

\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.

1. Use the graph to find the value of \(u\). [2]
2. Find the time taken for the particle to return to ground level. [3]

A small box of mass 5 kg is pulled at a constant speed of \(2.5 \text{ m s}^{-1}\) down a line of greatest slope of a rough plane inclined at \(10°\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

1. Find the coefficient of friction between the box and the plane. [5]

The pulling force is removed while the box is moving at \(2.5 \text{ m s}^{-1}\).

1. Find the distance moved by the box after the instant at which the pulling force is removed. [4]

\includegraphics{figure_2} The diagram shows a wire \(ABCD\) consisting of a straight part \(AB\) of length \(5\) m and a part \(BCD\) in the shape of a semicircle of radius \(6\) m and centre \(O\). The diameter \(BD\) of the semicircle is horizontal and \(AB\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(AB\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5\) m s\(^{-2}\) between \(A\) and \(B\).

1. Show that the speed of the ring as it reaches \(B\) is \(5\) m s\(^{-1}\). [1]

A particle \(A\) moves in a straight line with constant speed \(10\) m s\(^{-1}\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16\) m s\(^{-1}\) at \(O\) and has a constant deceleration of \(2\) m s\(^{-2}\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) s after \(B\) passes through \(O\). [3]
2. Find the distance between the particles when \(B\) comes to instantaneous rest. [3]
3. Find the minimum distance between the particles. [3]

\includegraphics{figure_6} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). A particle \(A\) of mass \(0.3\) kg rests on the horizontal surface 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 block. The other end of the string is attached to a particle \(B\) of mass \(1.5\) kg which rests on the sloping surface of the block. The system is released from rest with the string taut.

1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string. [5]
2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first \(3\) seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after \(3\) s is \(5\) m s\(^{-1}\). Find the acceleration of particle \(A\) and the value of \(\mu\). [9]

A particle \(P\) is projected vertically upwards with speed \(24 \text{ m s}^{-1}\) from a point \(5 \text{ m}\) above ground level. Find the time from projection until \(P\) reaches the ground. [3]

A sprinter runs a race of \(200 \text{ m}\). His total time for running the race is \(20 \text{ s}\). He starts from rest and accelerates uniformly for \(6 \text{ s}\), reaching a speed of \(12 \text{ m s}^{-1}\). He maintains this speed for the next \(10 \text{ s}\), before decelerating uniformly to cross the finishing line with speed \(V \text{ m s}^{-1}\).

1. Find the distance travelled by the sprinter in the first \(16 \text{ s}\) of the race. Hence sketch a displacement-time graph for the \(20 \text{ s}\) of the sprinter's race. [6]
2. Find the value of \(V\). [2]

\includegraphics{figure_1} The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of three straight line segments. The distance between the two stations is 9040 m.

1. Find the acceleration of the train during the first 40 s. [1]
2. Find the length of time for which the train is travelling at constant speed. [2]
3. Find the distance travelled by the train while it is decelerating. [2]

\includegraphics{figure_5} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(\frac{7}{3}\) m s\(^{-2}\). The velocity of the particle at \(t = 5\) is \(7\) m s\(^{-1}\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V\) m s\(^{-1}\).

1. Find the distance travelled by the particle in the first \(8\) s of its motion. [3]
2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is \(32\) m, find the value of \(V\). [4]

\includegraphics{figure_3} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V\) m s\(^{-1}\) at \(t = 10\).

1. Find the acceleration of the particle during the first \(2\) seconds. [1]
2. Find the value of \(V\). [2]

At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).

1. Find the distance \(AB\) and hence find the value of \(T\). [4]

\includegraphics{figure_2} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U\) m s\(^{-1}\). The tractor accelerates to a speed of \(V\) m s\(^{-1}\) over a period of 5 s, and then travels at this speed for a further 25 s. The tractor then accelerates to a speed of 12 m s\(^{-1}\) over a period of 5 s. The tractor then decelerates to rest over a period of 15 s.

1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is 0.8 m s\(^{-2}\), find the value of \(V\). [2]
2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m, find the value of \(U\). [3]

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 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]

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]

\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 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 train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.

1. Sketch a speed-time graph to show the motion of the train, [3]
2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]

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 \(t = T\) 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 racing car is travelling on a straight horizontal road. Its initial speed is \(25\) m s\(^{-1}\) and it accelerates for \(4\) s to reach a speed of \(V\) m s\(^{-1}\). It then travels at a constant speed of \(V\) m s\(^{-1}\) for a further \(8\) s. The total distance travelled by the car during this \(12\) s period is \(600\) m.

1. Sketch a speed-time graph to illustrate the motion of the car during this \(12\) s period. [2]
2. Find the value of \(V\). [4]
3. Find the acceleration of the car during the initial \(4\) s period. [2]

A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s, reaching a speed of 25 m s\(^{-1}\). The car then travels at a constant speed of 25 m s\(^{-1}\) for 120 s. Finally it moves with constant deceleration, coming to rest at a point \(F\).

1. In the space below, sketch a speed-time graph to illustrate the motion of the car. [2]

The distance between \(S\) and \(F\) is 4 km.

1. Calculate the total time the car takes to travel from \(S\) to \(F\). [3]

A motorcycle starts at \(S\), 10 s after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate

1. the time the motorcycle takes to travel from \(S\) to \(P\), [5]
2. the speed of the motorcycle at \(P\). [2]

A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
2. Find the time taken by the car to travel from \(P\) to \(S\). [3]