Multi-phase journey: find unknown speed or time

10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.

1. Sketch the velocity-time graph for Rory's run using this model.
2. Calculate \(T\).
3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
5. Sketch a velocity-time graph that represents Rory's run using this refined model.
6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)

6. A particle moving in a straight line with speed \(5 U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) undergoes a uniform deceleration for 6 seconds which reduces its speed to \(2 \mathrm { Um } \mathrm { s } ^ { - 1 }\). It maintains this speed for 16 seconds before uniformly decelerating to rest in a further 2 seconds.

1. Sketch a speed-time graph displaying this information.
2. Find an expression for each of the decelerations in terms of \(U\). Given that the total distance travelled by the particle during this period of motion is 220 m ,
3. find the value of \(U\).

5. A car on a straight test track starts from rest and accelerates to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 seconds. The car maintains this speed for a further 50 seconds before decelerating to rest. In a simple model of this motion, the acceleration and deceleration are assumed to be uniform and the magnitude of the deceleration to be 1.5 times that of the acceleration.

1. Show that the total time for which the car is moving is 60 seconds.
2. Sketch a velocity-time graph for this journey. Given that the total distance travelled is 1320 metres,
3. find \(V\). In a more sophisticated model, the acceleration is assumed to be inversely proportional to the velocity of the car.
4. Explain how the acceleration would vary during the first six seconds under this model.
   (2 marks)

7 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-07_512_1072_484_502} The diagram shows the velocity-time graph for a train travelling on a straight level track between stations \(A\) and \(B\) that are 2 km apart. The train leaves \(A\), accelerating uniformly from rest for 400 m until reaching a speed of \(32 \mathrm {~ms} ^ { - 1 }\). The train then travels at this steady speed for \(T\) seconds before decelerating uniformly at \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\). Find the total time for the journey.

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

In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. For this model of the motion of the train between \(A\) and \(B\),
   1. state the value of the constant velocity of the train,
   2. state the time for which the train is decelerating,
   3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.

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]