3.02c Interpret kinematic graphs: gradient and area

4 A car starts from rest and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 50 m . The car then travels with constant velocity for 500 m for a period of 25 s , before decelerating to rest. The magnitude of this deceleration is \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Sketch the velocity-time graph for the motion of the car. \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-05_533_1155_534_534}
2. Find the value of \(a\).
3. Find the total time for which the car is in motion.

6 An elevator is pulled vertically upwards by a cable. The elevator accelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s , then travels at constant speed for 25 s . The elevator then decelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) until it comes to rest.

1. Find the greatest speed of the elevator and hence draw a velocity-time graph for the motion of the elevator.
2. Find the total distance travelled by the elevator.
   The mass of the elevator is 1200 kg and there is a crate of mass \(m \mathrm {~kg}\) resting on the floor of the elevator.
3. Given that the tension in the cable when the elevator is decelerating is 12250 N , find the value of \(m\).
4. Find the greatest magnitude of the force exerted on the crate by the floor of the elevator, and state its direction.

1 A car starts from rest and accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) for 10 s . It then travels at a constant speed for 30 s . The car then uniformly decelerates to rest over a period of 20 s .

1. Sketch a velocity-time graph for the motion of the car. \includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-03_762_1081_447_493}
2. Find the total distance travelled by the car.

7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 , \\ s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$

1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
   At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-06_661_1529_260_306} An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph consists of 7 straight line segments. The elevator accelerates upwards from rest to a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 1.5 s and then travels at this speed for 4.5 s , before decelerating to rest over a period of 1 s . The elevator then remains at rest for 6 s , before accelerating to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards over a period of 2 s . The elevator travels at this speed for a period of 5 s , before decelerating to rest over a period of 1.5 s .

1. Find the acceleration of the elevator during the first 1.5 s .
2. Given that the elevator starts and finishes its journey on the ground floor, find \(V\).
3. The combined weight of the elevator and passengers on its upward journey is 1500 kg . Assuming that there is no resistance to motion, find the tension in the elevator cable on its upward journey when the elevator is decelerating.

4
The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at a constant rate of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(T \mathrm {~s}\). It then decelerates at a constant rate for 5 seconds before travelling at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 27.5 s . The car then decelerates to rest at a constant rate over a period of 5 s .

1. Find \(T\).
2. Given that the distance travelled up to the point at which the car begins to move with constant speed is one third of the total distance travelled, find \(V\).

1 A bus moves from rest with constant acceleration for 12 s . It then moves with constant speed for 30 s before decelerating uniformly to rest in a further 6 s . The total distance travelled is 585 m .

1. Find the constant speed of the bus.
2. Find the magnitude of the deceleration.

1 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-02_611_1351_260_397} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of six straight line segments. The car accelerates from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s , and then travels at this speed for a further 20 s . The car then decelerates to a speed of \(6 \mathrm {~ms} ^ { - 1 }\) over a period of 5 s . This speed is maintained for a further \(( T - 30 ) \mathrm { s }\). The car then accelerates again to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(( 50 - T ) \mathrm { s }\), before decelerating to rest over a period of 10 s .

1. Given that during the two stages of the motion when the car is accelerating, the accelerations are equal, find the value of \(T\).
2. Find the total distance travelled by the car during the motion.

3 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-04_666_1278_280_424} The diagram shows the velocity-time graph for the motion of a bus. The bus starts from rest and accelerates uniformly for 8 seconds until it reaches a speed of \(12.6 \mathrm {~ms} ^ { - 1 }\). The bus maintains this speed for 40 seconds. It then decelerates uniformly in two stages. Between 48 and 62 seconds the bus decelerates at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and between 62 and 70 seconds it decelerates at \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until coming to rest.

1. Find the distance covered by the bus in the first 8 seconds.
2. Find the value of \(a\).
3. Find the average speed of the bus for the whole journey.

3 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_556_974_1548_587} The diagram shows the velocity-time graphs for the motion of two cyclists \(P\) and \(Q\), who travel in the same direction along a straight path. Both cyclists start from rest at the same point \(O\) and both accelerate at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Both then continue at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(Q\) starts his journey \(T\) seconds after \(P\).

1. Show in a sketch of the diagram the region whose area represents the displacement of \(P\), from \(O\), at the instant when \(Q\) starts. Given that \(P\) has travelled 16 m at the instant when \(Q\) starts, find
2. the value of \(T\),
3. the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(10 \mathrm {~ms} ^ { - 1 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-3_727_899_267_625} A boy runs from a point \(A\) to a point \(C\). He pauses at \(C\) and then walks back towards \(A\) until reaching the point \(B\), where he stops. The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the boy's velocity at time \(t\) seconds after leaving \(A\). The boy runs and walks in the same straight line throughout.

1. Find the distances \(A C\) and \(A B\).
2. Sketch the graph of \(x\) against \(t\), where \(x\) metres is the boy's displacement from \(A\). Show clearly the values of \(t\) and \(x\) when the boy arrives at \(C\), when he leaves \(C\), and when he arrives at \(B\). [3]

6 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_593_746_269_701} A particle \(P\) starts from rest at the point \(A\) and travels in a straight line, coming to rest again after 10 s . The velocity-time graph for \(P\) consists of two straight line segments (see diagram). A particle \(Q\) starts from rest at \(A\) at the same instant as \(P\) and travels along the same straight line as \(P\). The velocity of \(Q\) is given by \(v = 3 t - 0.3 t ^ { 2 }\) for \(0 \leqslant t \leqslant 10\). The displacements from \(A\) of \(P\) and \(Q\) are the same when \(t = 10\).

1. Show that the greatest velocity of \(P\) during its motion is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(t\), in the interval \(0 < t < 5\), for which the acceleration of \(Q\) is the same as the acceleration of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543} An object \(P\) travels from \(A\) to \(B\) in a time of 80 s . The diagram shows the graph of \(v\) against \(t\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(A\). The graph consists of straight line segments for the intervals \(0 \leqslant t \leqslant 10\) and \(30 \leqslant t \leqslant 80\), and a curved section whose equation is \(v = - 0.01 t ^ { 2 } + 0.5 t - 1\) for \(10 \leqslant t \leqslant 30\). Find

1. the maximum velocity of \(P\),
2. the distance \(A B\).

5 A ball moves on the horizontal surface of a billiards table with deceleration of constant magnitude \(d \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball starts at \(A\) with speed \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and reaches the edge of the table at \(B , 1.2 \mathrm {~s}\) later, with speed \(1.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the distance \(A B\) and the value of \(d\). \(A B\) is at right angles to the edge of the table containing \(B\). The table has a low wall along each of its edges and the ball rebounds from the wall at \(B\) and moves directly towards \(A\). The ball comes to rest at \(C\) where the distance \(B C\) is 2 m .
2. Find the speed with which the ball starts to move towards \(A\) and the time taken for the ball to travel from \(B\) to \(C\).
3. Sketch a velocity-time graph for the motion of the ball, from the time the ball leaves \(A\) until it comes to rest at \(C\), showing on the axes the values of the velocity and the time when the ball is at \(A\), at \(B\) and at \(C\).

5 A train starts from rest at a station \(A\) and travels in a straight line to station \(B\), where it comes to rest. The train moves with constant acceleration \(0.025 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 600 s , with constant speed for the next 2600 s , and finally with constant deceleration \(0.0375 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the total time taken for the train to travel from \(A\) to \(B\).
2. Sketch the velocity-time graph for the journey and find the distance \(A B\).
3. The speed of the train \(t\) seconds after leaving \(A\) is \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). State the possible values of \(t\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-2_748_1410_979_370} The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:

• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.
  1. Show that the total distance fallen is 1048 m .

The weight of the parachutist is 850 N .

Find the upward force on the parachutist due to the parachute, during the second stage.

3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .

1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.

1 A bus moves in a straight line between two bus stops. The bus starts from rest and accelerates at \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . The bus then travels for 24 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 6 s . Sketch a velocity-time graph for the motion and hence find the distance between the two bus stops.

2 \includegraphics[max width=\textwidth, alt={}, center]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-2_649_1244_482_452} A man runs in a straight line. He passes through a fixed point \(A\) with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). At time \(t \mathrm {~s}\) his velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows the graph of \(v\) against \(t\) for the period \(0 \leqslant t \leqslant 40\).

1. Show that the man runs more than 154 m in the first 24 s .
2. Given that the man runs 20 m in the interval \(20 \leqslant t \leqslant 24\), find how far he is from \(A\) when \(t = 40\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-4_547_1237_269_456} A tractor \(A\) starts from rest and travels along a straight road for 500 seconds. The velocity-time graph for the journey is shown above. This graph consists of three straight line segments. Find

1. the distance travelled by \(A\),
2. the initial acceleration of \(A\). Another tractor \(B\) starts from rest at the same instant as \(A\), and travels along the same road for 500 seconds. Its velocity \(t\) seconds after starting is \(\left( 0.06 t - 0.00012 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
3. how much greater \(B\) 's initial acceleration is than \(A\) 's,
4. how much further \(B\) has travelled than \(A\), at the instant when \(B\) 's velocity reaches its maximum.

5 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).

1. Sketch the velocity-time graph for the journey.
2. Find the velocity at \(t = 80\).
3. Find the total distance for the journey.
4. Find the acceleration of the car for \(0 < t < 80\).

6 A train travels from \(A\) to \(B\), a distance of 20000 m , taking 1000 s . The journey has three stages. In the first stage the train starts from rest at \(A\) and accelerates uniformly until its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the second stage the train travels at constant speed \(V _ { \mathrm { m } } { } ^ { - 1 }\) for 600 s . During the third stage of the journey the train decelerates uniformly, coming to rest at \(B\).

1. Sketch the velocity-time graph for the train's journey.
2. Find the value of \(V\).
3. Given that the acceleration of the train during the first stage of the journey is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the distance travelled by the train during the third stage of the journey. \(7 \quad\) A particle \(P\) is held at rest at a fixed point \(O\) and then released. \(P\) falls freely under gravity until it reaches the point \(A\) which is 1.25 m below \(O\).
4. Find the speed of \(P\) at \(A\) and the time taken for \(P\) to reach \(A\). The particle continues to fall, but now its downward acceleration \(t\) seconds after passing through \(A\) is \(( 10 - 0.3 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
5. Find the total distance \(P\) has fallen, 3 s after being released from \(O\).

6 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).

1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
2. the value of \(V\),
3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
4. the deceleration of \(P\) immediately before it reaches \(B\). \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).

4 A particle starts from rest at a point \(X\) and moves in a straight line until, 60 seconds later, it reaches a point \(Y\). At time \(t \mathrm {~s}\) after leaving \(X\), the acceleration of the particle is $$\begin{array} { r c c } 0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 0 < t < 4 \\ 0 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 4 < t < 54 \\ - 0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 } & \text { for } & 54 < t < 60 \end{array}$$

1. Find the velocity of the particle when \(t = 4\) and when \(t = 60\), and sketch the velocity-time graph.
2. Find the distance \(X Y\).

7 A tractor travels in a straight line from a point \(A\) to a point \(B\). The velocity of the tractor is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\).

1. \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-4_668_1091_397_568} The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for
   1. the distance \(A B\),
   2. the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).
   3. The actual velocity of the tractor is given by \(v = 0.04 t - 0.00005 t ^ { 2 }\) for \(0 \leqslant t \leqslant 800\).
      (a) Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i). For the interval \(0 \leqslant t \leqslant 400\), the approximate velocity of the tractor in part (i) is denoted by \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (b) Express \(v _ { 1 }\) in terms of \(t\) and hence show that \(v _ { 1 } - v = 0.00005 ( t - 200 ) ^ { 2 } - 1\).
   4. Deduce that \(- 1 \leqslant v _ { 1 } - v \leqslant 1\).