3.02b Kinematic graphs: displacement-time and velocity-time

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 \(V\).

3. A car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The car is modelled as a particle.
At time \(t = 0\), the car is at the point \(A\) and the driver sees a road sign 48 m ahead.
Let \(t\) seconds be the time that elapses after the car passes \(A\).
In a first model, the car is assumed to decelerate uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) from \(A\) until the car reaches the road sign.

1. Use this first model to find the speed of the car as it reaches the sign. The road sign indicates that the speed limit immediately after the sign is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   In a second model, the car is assumed to decelerate uniformly at \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(A\) until it reaches a speed of \(13 \mathrm {~ms} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
2. Use this second model to find the value of \(t\) at which the car reaches the sign. In a third model, the car is assumed to move with constant speed \(25 \mathrm {~ms} ^ { - 1 }\) from \(A\) until time \(t = 0.2\), the car then decelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
3. Use this third model to find the value of \(t\) at which the car reaches the sign.

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} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 3.5 s , reaching a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Car \(A\) then moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(a\). Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with a constant acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\).
3. Find the value of \(T\).
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\).
5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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3. A car moves along a horizontal straight road, passing two points \(A\) and \(B\). At \(A\) the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the driver passes \(A\), he sees a warning sign \(W\) ahead of him, 120 m away. He immediately applies the brakes and the car decelerates with uniform deceleration, reaching \(W\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(W\), the driver sees that the road is clear. He then immediately accelerates the car with uniform acceleration for 16 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 } ( V > 15 )\). He then maintains the car at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Moving at this constant speed, the car passes \(B\) after a further 22 s .

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from \(A\) to \(B\).
2. Find the time taken for the car to move from \(A\) to \(B\). The distance from \(A\) to \(B\) is 1 km .
3. Find the value of \(V\).

2. A small ball is projected vertically upwards from ground level with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball takes 4 s to return to ground level.

1. Draw, in the space below, a velocity-time graph to represent the motion of the ball during the first 4 s .
2. The maximum height of the ball above the ground during the first 4 s is 19.6 m . Find the value of \(u\).

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 \(t = 400\).

3 A man travels 360 m along a straight road. He walks for the first 120 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), runs the next 180 m at \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then walks the final 60 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \(( t , x )\) graph for the journey, showing the values of \(t\) for which \(x = 120,300\) and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
2. Draw a dotted line on your ( \(t , x\) ) graph to represent the woman's journey.
3. Calculate the value of \(t\) at which the man overtakes the woman.

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.

1 A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the particle in the interval \(0 < t < 6\).
2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
3. When \(t = 0\) the particle is at A . Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\).

1 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 \(V\).

1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h] \end{figure}

1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
3. State the direction of motion of the ring when
   (A) \(t = 1\),
   (B) \(t = 2.75\),
   (C) \(t = 3.25\).

1 An object C is moving along a vertical straight line. Fig. 1 shows the velocity-time graph for part of its motion. Initially C is moving upwards at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after 10 s it is moving downwards at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h] \end{figure} C then moves as follows.

• In the interval \(10 \leqslant t \leqslant 15\), the velocity of C is constant at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
• In the interval \(15 \leqslant t \leqslant 20\), the velocity of C increases uniformly so that C has zero velocity at \(t = 20\).
  1. Complete the velocity-time graph for the motion of C in the time interval \(0 \leqslant t \leqslant 20\).
  2. Calculate the acceleration of C in the time interval \(0 < t < 10\).
  3. Calculate the displacement of C from \(t = 0\) to \(t = 20\).

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 .

6. \includegraphics[max width=\textwidth, alt={}, center]{14a2477a-c40e-4b4b-bc39-7100d1df9b4d-2_397_488_1299_632} The diagram shows a vertical cross-section through a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60 ^ { \circ }\). When the depth of water in the vase is \(h \mathrm {~cm}\), the volume of water in the vase is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\). The vase is initially empty and water is poured in at a constant rate of \(120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
2. Find, to 2 decimal places, the rate at which \(h\) is increasing
   1. when \(h = 6\),
   2. after water has been poured in for 8 seconds.

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 \(0 \leqslant t \leqslant 18\).

4. A particle \(P\) moves along the \(x\)-axis in a straight line so that, at time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = \begin{cases} 10 t - 2 t ^ { 2 } , & 0 \leqslant t \leqslant 6 \\ \frac { - 432 } { t ^ { 2 } } , & t > 6 \end{cases}$$ At \(t = 0 , P\) is at the origin \(O\). Find the displacement of \(P\) from \(O\) when

1. \(t = 6\),
2. \(t = 10\).

3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find

1. the value of \(c\) ,
2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D啨
   (b)the acceleration of \(P\) when \(t = 1.5\) .

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.