Multi-stage motion with algebraic unknowns

A tram starts from rest and moves with uniform acceleration for 20 s. The tram then travels at a constant speed, \(V \text{ ms}^{-1}\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km.

1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving. [2]
2. Find \(V\). [2]
3. Find the magnitude of the acceleration. [2]

A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \text{ ms}^{-2}\) until it reaches a speed of \(16 \text{ ms}^{-1}\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 \text{ ms}^{-2}\), coming to rest at \(B\). The total time for the journey is \(240\) s.

1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\). [1]
2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed. [2]
3. Given that the distance from \(A\) to \(B\) is \(3000\) m, find the value of \(a\). [3]

A particle moves in a straight line. It starts from rest, at time \(t = 0\), and accelerates at 0.6 t ms\(^{-2}\) for 4 s, reaching a speed of \(V\) ms\(^{-1}\). The particle then travels at \(V\) ms\(^{-1}\) for 11 s, and finally slows down, with constant deceleration, stopping after a further 5 s.

1. Show that \(V = 4.8\). [1]
2. Sketch a velocity-time graph for the motion. [3]
3. Find an expression, in terms of \(t\), for the velocity of the particle for \(15 \leqslant t \leqslant 20\). [2]
4. Find the total distance travelled by the particle. [4]

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 until it reaches its maximum speed of 108 km h\(^{-1}\). The train then travels at this speed before it moves with constant deceleration coming to rest at \(B\). The journey from \(A\) to \(B\) takes 8 minutes.

1. Change 108 km h\(^{-1}\) into m s\(^{-1}\). [2]
2. Sketch a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). [2]

Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,

1. find the acceleration, in m s\(^{-2}\), of the train. [6]

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 car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is \(22 \text{ m s}^{-1}\). The car maintains this speed for \(T\) seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.

1. Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights. [2]
2. Find the value of \(T\). [3]

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, \(a \text{ m s}^{-2}\), and passes the car at the point \(A\) which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed \(22 \text{ m s}^{-1}\).

1. Find the time it takes for the motorcycle to move from the first set of traffic lights to the point \(A\). [4]
2. Find the value of \(a\). [2]

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 train starts from rest at a station \(S\) and accelerates at a constant rate for \(2x\) seconds to a speed of \(5x\) ms\(^{-1}\). It maintains this speed until 126 seconds after it left \(S\) and then decelerates at a constant rate until it comes to rest at another station \(T\), \(20x\) seconds after it left \(S\).

1. Sketch a velocity-time graph for this journey. [4 marks]

Given that the distance between \(S\) and \(T\) is \(5.4\) km,

1. show that \(x^2 + 7x = 120\). [4 marks]
2. Find the value of \(x\). [3 marks]

A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).

1. Sketch the velocity-time graph for the jogger's journey. [2]
2. Show that \(3V^2 - 100V + 352 = 0\). [6]
3. Hence find the value of \(V\), giving a reason for your answer. [3]