Distance from velocity-time graph

3 The diagram shows a velocity-time graph for a train as it moves on a straight horizontal track for 50 seconds. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-3_620_1221_408_358}

1. Find the distance that the train moves in the first 28 seconds.
2. Calculate the total distance moved by the train during the 50 seconds.
3. Hence calculate the average speed of the train.
4. Find the displacement of the train from its initial position when it has been moving for 50 seconds.
5. Hence calculate the average velocity of the train.
6. Find the acceleration of the train in the first 18 seconds of its motion.

2 A train travels along a straight horizontal track between two points \(A\) and \(B\).
Initially the train is at \(A\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Due to a problem, the train has to slow down and stop. At time \(t = 40\) seconds it begins to move again. At time \(t = 120\) seconds the train is at \(B\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) again. The graph below shows how the velocity of the train varies as it moves from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-2_408_1086_1505_434}

1. Use the graph to find the total distance between the points \(A\) and \(B\).
2. The train should have travelled between \(A\) and \(B\) at a constant velocity of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the time that the train would take to travel between \(A\) and \(B\) at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Calculate the time by which the train was delayed.
3. The train has mass 500 tonnes. Find the resultant force acting on the train when \(40 < t < 120\).
   (4 marks)

1 A bus slows down as it approaches a bus stop. It stops at the bus stop and remains at rest for a short time as the passengers get on. It then accelerates away from the bus stop. The graph shows how the velocity of the bus varies. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-02_627_1296_657_402} Assume that the bus travels in a straight line during the motion described by the graph.

1. State the length of time for which the bus is at rest.
2. Find the distance travelled by the bus in the first 40 seconds.
3. Find the total distance travelled by the bus in the 120 -second period.
4. Find the average speed of the bus in the 120 -second period.
5. If the bus had not stopped but had travelled at a constant \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the 120 -second period, how much further would it have travelled?
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-03_2484_1709_223_153}

1 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h] \end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.

4 The velocity-time graph shown in Fig. 1 represents the straight line motion of a toy car. All the lines on the graph are straight. \begin{figure}[h] \end{figure} The car starts at the point A at \(t = 0\) and in the next 8 seconds moves to a point B .

1. Find the distance from A to B . \(T\) seconds after leaving A , the car is at a point C which is a distance of 10 m from B .
2. Find the value of \(T\).
3. Find the displacement from A to C .

4 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) at time \(t\) seconds. \begin{figure}[h] \end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).

3 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h] \end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.

14 A motorised scooter is travelling along a straight path with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over time \(t\) seconds as shown by the following graph. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-20_1120_1134_420_452} Noosha says that, in the period \(\mathbf { 1 2 } \leq \boldsymbol { t } \leq \mathbf { 3 6 }\), the scooter travels approximately 130 metres. Determine if Noosha is correct, showing clearly any calculations you have used.

15 A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) changes with time, \(t\) seconds. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-23_509_746_456_648} Over the period \(0 \leq t \leq 15\) the car has a total displacement of - 7 metres.
Initially the car has velocity \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the next time when the velocity of the car is \(0 \mathrm {~ms} ^ { - 1 }\) [0pt] [4 marks]

12 A particle moves in a straight line.
After the first 4 seconds of its motion, the displacement of the particle from its initial position is 0 metres. One of the graphs on the opposite page shows the velocity \(v \mathrm {~ms} ^ { - 1 }\) of the particle after time \(t\) seconds of its motion. Identify the correct graph.
Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-19_2249_896_260_484}

6. \begin{figure}[h] \end{figure} A car moves along a straight horizontal road. At time \(t = 0\), the velocity of the car is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then accelerates with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds. The car travels a distance \(D\) metres during these \(T\) seconds. Figure 1 shows the velocity-time graph for the motion of the car for \(0 \leqslant t \leqslant T\).
Using the graph, show that \(D = U T + 1 / 2 a T ^ { 2 }\).
(No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.)

\includegraphics{figure_2} A particle starts from the point \(A\) and travels in a straight line. The diagram shows the \((t, v)\) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leq t \leq 290\).

1. Find the value of \(t\) for which the distance of the particle from \(A\) is greatest. [2]
2. Find the displacement of the particle from \(A\) when \(t = 290\). [3]
3. Find the total distance travelled by the particle during the interval \(0 \leq t \leq 290\). [2]

Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v\text{ m s}^{-1}\) at time \(t\) seconds. \includegraphics{figure_1} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m. Find the value of \(V\). [4]

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point \(A\) when \(t = 0\) The mass of the train is 800kg. \includegraphics{figure_14}

1. Find the total distance travelled in the 20 seconds. [3 marks]
2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds. [1 mark]
3. Find the maximum magnitude of the resultant force acting on the train. [2 marks]
4. Explain why, in reality, the graph may not be an accurate model of the motion of the train. [1 mark]

The diagram below shows the velocity-time graph of a car moving along a straight road, where \(v\) m s\(^{-1}\) is the velocity of the car at time \(t\) s after it passes through the point \(A\). \includegraphics{figure_9}

1. Calculate the acceleration of the car at \(t = 6\). [2]
2. Jasmit says "The distance travelled by the car during the first 20 seconds of the car's motion is more than five times its displacement from \(A\) after the first 20 seconds of the car's motion". Give evidence to support Jasmit's statement. [3]