3.02c Interpret kinematic graphs: gradient and area

7. Two small children, Ajaz and Beth, are running a 100 m race along a straight horizontal track. They both start from rest, leaving the start line at the same time. Ajaz accelerates at \(0.8 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains this speed until he crosses the finish line. Beth accelerates at \(1 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds and then maintains a constant speed until she crosses the finish line. Ajaz and Beth cross the finish line at the same time.

1. Sketch, on the same axes, a speed-time graph for each child, from the instant when they leave the start line to the instant when they cross the finish line.
2. Find the time taken by Ajaz to complete the race.
3. Find the value of \(T\)
4. Find the difference in the speeds of the two children as they cross the finish line.

5. \begin{figure}[h] \end{figure} The speed-time graph in Figure 2 illustrates the motion of a car travelling along a straight horizontal road.
At time \(t = 0\), the car starts from rest and accelerates uniformly for 30 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until time \(t = T\) seconds.

1. Show that the distance travelled by the car between \(t = 0\) and \(t = T\) seconds is \(V ( T - 15 )\) metres. A motorbike also travels along the same road.
   At time \(t = T\) seconds, the distance travelled by each vehicle is the same.
2. Find the value of \(T\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-15_643_1266_1882_402} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}

1. A parachute is used to deliver a box of supplies. The parachute is attached to the box.

• the parachute and box are dropped from rest from a helicopter that is hovering at a height of 520 m above the ground
• the parachute and box fall vertically and freely under gravity for 5 seconds, then the parachute opens
• from the instant the parachute opens, it provides a resistance to motion of magnitude 3200 N
• the parachute and box continue to fall vertically downwards after the parachute opens
• the parachute and box are modelled throughout the motion as a particle \(P\) of mass 250 kg
  1. Find the distance fallen by \(P\) in the first 5 seconds.
  2. Find the speed with which \(P\) lands on the ground.
  3. Find the total time from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.
  4. Sketch a speed-time graph for the motion of \(P\) from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.

7. A train moves on a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(1 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train maintains this speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next \(T\) seconds before slowing down with constant deceleration \(0.5 \mathrm {~ms} ^ { - 2 }\), coming to rest at \(B\). The journey from \(A\) to \(B\) takes 180 s and the distance between the stations is 4800 m .

1. Sketch a speed-time graph for the motion of the train from \(A\) to \(B\).
2. Show that \(T = 180 - 3 V\).
3. Find the value of \(V\).

6. \begin{figure}[h] \end{figure} A small ball is thrown vertically upwards at time \(t = 0\) from a point \(A\) which is above horizontal ground. The ball hits the ground 7 s later. The ball is modelled as a particle moving freely under gravity.
The velocity-time graph shown in Figure 3 represents the motion of the ball for \(0 \leqslant t \leqslant 7\)

1. Find the speed with which the ball is thrown.
2. Find the height of \(A\) above the ground.

2. \begin{figure}[h] \end{figure} Two fixed points, \(A\) and \(B\), are on a straight horizontal road.
The acceleration-time graph in Figure 2 represents the motion of a car travelling along the road as it moves from \(A\) to \(B\). At time \(t = 0\), the car passes through \(A\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) At time \(t = 20 \mathrm {~s}\), the car passes through \(B\) with speed \(v \mathrm {~ms} ^ { - 1 }\)

1. Show that \(v = 18\)
2. Sketch a speed-time graph for the motion of the car from \(A\) to \(B\).
3. Find the distance \(A B\).

6 \begin{figure}[h] \end{figure} A cyclist \(P\) travels along a straight road starting from rest at \(A\) and accelerating at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(B 20 \mathrm {~s}\) after leaving \(A\). Fig. 1 shows the ( \(t , v\) ) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\). Find

1. the time for which \(P\) is accelerating,
2. the distance \(A B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_607_937_1420_605} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Another cyclist \(Q\) travels along the same straight road in the opposite direction. She starts at rest from \(B\) at the same instant that \(P\) leaves \(A\). Cyclist \(Q\) accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(8 \mathrm {~ms} ^ { - 1 }\) and continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(A 20 \mathrm {~s}\) after leaving \(B\). Fig. 2 shows the \(( t , x )\) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\), where \(x\) is the displacement of \(Q\) from \(A\) towards \(B\).
3. Sketch a copy of Fig. 1 and add to your copy a sketch of the ( \(t , v\) ) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\).
4. Sketch a copy of Fig. 2 and add to your copy a sketch of the \(( t , x )\) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\).
5. Find the value \(t\) at the instant that \(P\) and \(Q\) pass each other. \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-5_447_739_269_703} The upper edge of a smooth plane inclined at \(70 ^ { \circ }\) to the horizontal is joined to an edge of a rough horizontal table. Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth pulley which is fixed at the top of the smooth inclined plane. Particle \(A\) is held in contact with the rough horizontal table and particle \(B\) is in contact with the smooth inclined plane with the string taut (see diagram). The coefficient of friction between \(A\) and the horizontal table is 0.4 . Particle \(A\) is released from rest and the system starts to move.
6. Find the acceleration of \(A\) and the tension in the string. The string breaks when the speed of the particles is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
7. Assuming \(A\) does not reach the pulley, find the distance travelled by \(A\) after the string breaks.
8. Assuming \(B\) does not reach the ground before \(A\) stops, find the distance travelled by \(B\) from the time the string breaks to the time that \(A\) stops.

5 A car is travelling at \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight road when it passes a point \(A\) at time \(t = 0\), where \(t\) is in seconds. For \(0 \leqslant t \leqslant 6\), the car accelerates at \(0.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the speed of the car when \(t = 6\).
2. Calculate the displacement of the car from \(A\) when \(t = 6\).
3. Three \(( t , x )\) graphs are shown below, for \(0 \leqslant t \leqslant 6\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_458_1366_340} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_460_1366_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_384_461_1366_1420} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
   1. State which of these three graphs is most appropriate to represent the motion of the car.
   2. For each of the two other graphs give a reason why it is not appropriate to represent the motion of the car.

6 Small parcels are being loaded onto a trolley. Initially the parcels are 2.5 m above the trolley.

1. A parcel is released from rest and falls vertically onto the trolley. Calculate
   1. the time taken for a parcel to fall onto the trolley,
   2. the speed of a parcel when it strikes the trolley.
   3. \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_327_723_603_751} Parcels are often damaged when loaded in the way described, so a ramp is constructed down which parcels can slide onto the trolley. The ramp makes an angle of \(60 ^ { \circ }\) to the vertical, and the coefficient of friction between the ramp and a parcel is 0.2 . A parcel of mass 2 kg is released from rest at the top of the ramp (see diagram). Calculate the speed of the parcel after sliding down the ramp.

7 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   1. sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   2. calculate the distance between the two particles at the instant when they are projected.

7 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-4_634_1127_934_507} A car \(P\) starts from rest and travels along a straight road for 600 s . The \(( t , v )\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

1. the distance travelled by \(P\),
2. the deceleration of \(P\) during the interval \(500 < t < 600\). Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for 600 s . At time \(t \mathrm {~s}\) after starting the velocity of \(Q\) is \(\left( 600 t ^ { 2 } - t ^ { 3 } \right) \times 10 ^ { - 6 } \mathrm {~ms} ^ { - 1 }\).
3. Find an expression in terms of \(t\) for the acceleration of \(Q\).
4. Find how much less \(Q\) 's deceleration is than \(P\) 's when \(t = 550\).
5. Show that \(Q\) has its maximum velocity when \(t = 400\).
6. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\).

2 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_714_1048_1231_552} A particle starts from the point A and travels in a straight line. The diagram shows the ( \(\mathrm { t } , \mathrm { v }\) ) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leqslant t \leqslant 290\).

1. Find the value of ther which the distance of the particle from A is greatest.
2. Find the displacement of the particle from A when \(\mathrm { t } = 290\).
3. Find the total distance travelled by the particle during the interval \(0 \leqslant \mathrm { t } \leqslant 290\).

5 \includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.

1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.

6 The speed of a 100 metre runner in \(\mathrm { ms } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 }$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time?
5. Find the greatest acceleration of the runner according to each model.

1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h] \end{figure}

1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.

6 A toy car is travelling in a straight horizontal line.
One model of the motion for \(0 \leqslant t \leqslant 8\), where \(t\) is the time in seconds, is shown in the velocity-time graph Fig. 6. \begin{figure}[h] \end{figure}

1. Calculate the distance travelled by the car from \(t = 0\) to \(t = 8\).
2. How much less time would the car have taken to travel this distance if it had maintained its initial speed throughout?
3. What is the acceleration of the car when \(t = 1\) ? From \(t = 8\) to \(t = 14\), the car travels 58.5 m with a new constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Find \(a\). A second model for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the toy car is $$v = 12 - 10 t + \frac { 9 } { 4 } t ^ { 2 } - \frac { 1 } { 8 } t ^ { 3 } , \text { for } 0 \leqslant t \leqslant 8$$ This model agrees with the values for \(v\) given in Fig. 6 for \(t = 0,2,4\) and 6. [Note that you are not required to verify this.] Use this second model to answer the following questions.
5. Calculate the acceleration of the car when \(t = 1\).
6. Initially the car is at A. Find an expression in terms of \(t\) for the displacement of the car from A after the first \(t\) seconds of its motion. Hence find the displacement of the car from A when \(t = 8\).
7. Explain with a reason what this model predicts for the motion of the car between \(t = 2\) and \(t = 4\).

2 A car passes a point A travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Its motion over the next 45 seconds is modelled as follows.

• The car's speed increases uniformly from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the first 10 s .
• Its speed then increases uniformly to \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the next 15 s .
• The car then maintains this speed for a further 20 s at which time it reaches the point B .
  1. Sketch a speed-time graph to represent this motion.
  2. Calculate the distance from A to B .
  3. When it reaches the point B , the car is brought uniformly to rest in \(T\) seconds. The total distance from A is now 1700 m . Calculate the value of \(T\).

1 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 .

1 Fig. 1 shows the velocity-time graph of a cyclist travelling along a straight horizontal road between two sets of traffic lights. The velocity, \(v\), is measured in metres per second and the time, \(t\), in seconds. The distance travelled, \(s\) metres, is measured from when \(t = 0\). \begin{figure}[h] \end{figure}

1. Find the values of \(s\) when \(t = 4\) and when \(t = 18\).
2. Sketch the graph of \(s\) against \(t\) for \(0 \leqslant t \leqslant 18\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v\) is given by

$$v = \left\{ \begin{array} { l c } 8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\ 16 - 2 t , & t > 4 . \end{array} \right.$$ When \(t = 0 , P\) is at the origin \(O\).
Find

1. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\),
2. the distance of \(P\) from \(O\) when \(t = 4\),
3. the time at which \(P\) is instantaneously at rest for \(t > 4\),
4. the total distance travelled by \(P\) in the first 10 s of its motion.

2. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 8 t - t ^ { 2 }$$

1. Find the maximum value of \(v\).
2. Find the time taken for \(P\) to return to \(O\).

3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find

1. the times when \(P\) is instantaneously at rest,
2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

4 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-2_657_1495_1539_324} A car travelling on a straight road accelerates from rest to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . It continues at constant speed for 11 s and then decelerates to rest in 2 s . The driver gets out of the car and walks at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 20 s back to a shop which he enters. Some time later he leaves the shop and jogs to the car at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He arrives at the vehicle 60 s after it began to accelerate from rest. The diagram, which has six straight line segments, shows the \(( t , v )\) graph for the motion of the driver.

1. Calculate the initial acceleration and final deceleration of the car.
2. Calculate the distance the car travels.
3. Calculate the length of time the driver is in the shop.

3 \(\mathrm { v } \left( \mathrm { ms } ^ { - 1 } \right)\) \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_449_1121_1500_440}
not to scale The diagram shows the \(( t , v )\) graphs for two athletes, \(A\) and \(B\), who run in the same direction in the same straight line while they exchange the baton in a relay race. \(A\) runs with constant velocity \(10 \mathrm {~ms} ^ { - 1 }\) until he decelerates at \(5 \mathrm {~ms} ^ { - 2 }\) and subsequently comes to rest. \(B\) has constant acceleration from rest until reaching his constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). The baton is exchanged 2 s after \(B\) starts running, when both athletes have speed \(8 \mathrm {~ms} ^ { - 1 }\) and \(B\) is 1 m ahead of \(A\).

1. Find the value of \(t\) at which \(A\) starts to decelerate.
2. Calculate the distance between \(A\) and \(B\) at the instant when \(B\) starts to run.

4 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-3_298_540_262_735} The diagram shows the \(( t , v )\) graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\). The car passes through \(A\) with velocity \(18 \mathrm {~ms} ^ { - 1 }\), and moves with constant acceleration \(2.4 \mathrm {~ms} ^ { - 2 }\) until \(t = 5\). The car subsequently moves with constant velocity until it is 300 m from \(A\). When the car is more than 300 m from \(A\), it has constant deceleration \(6 \mathrm {~ms} ^ { - 2 }\), until it comes to rest.

1. Find the greatest speed of the car.
2. Calculate the value of \(t\) for the instant when the car begins to decelerate.
3. Calculate the distance from \(A\) of the car when it is at rest.