Travel graphs

2. \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-04_269_1175_296_375}

1. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{3a8395fd-6e44-48a1-8c97-3365a284956a-02_404_755_312_577} Figure 1 shows the speed-time graph of a cyclist moving on a straight road over a 7 s period. The sections of the graph from \(t = 0\) to \(t = 3\), and from \(t = 3\) to \(t = 7\), are straight lines. The section from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis. State what can be deduced about the motion of the cyclist from the fact that

1. the graph from \(t = 0\) to \(t = 3\) is a straight line,
2. the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
3. Find the distance travelled by the cyclist during this 7 s period.

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]

The graph shows how the speed of a cyclist varies during a timed section of length 120 metres along a straight track. \includegraphics{figure_15}

1. Find the acceleration of the cyclist during the first 10 seconds. [1 mark]
2. After the first 15 seconds, the cyclist travels at a constant speed of 5 m s⁻¹ for a further \(T\) seconds to complete the 120-metre section. Calculate the value of \(T\). [4 marks]

\includegraphics{figure_2} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find

1. the acceleration of the tool during the first 2 s of the motion, [1]
2. the distance the tool moves forward while cutting, [2]
3. the greatest speed of the tool during the return to its starting position. [2]

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 \(P\).

A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}

A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}

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 \(0 < t < 80\).

10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }

10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }

\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]

A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \text{ m s}^{-2}\) until it passes the point \(A\) when it is moving with speed \(10 \text{ m s}^{-1}\). It then moves with constant acceleration \(3 \text{ m s}^{-2}\) for 6 s until it reaches the point \(B\). Find

1. the speed of the car at \(B\), [2]
2. the distance \(OB\). [5]

An aircraft moves along a straight horizontal runway with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\). Points \(A\) and \(B\) lie on the runway. The aircraft passes \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and its speed at \(B\) must be at least \(78 \mathrm {~ms} ^ { - 1 }\) if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes \(A\).
b) Determine the minimum value of the distance \(A B\) for the aircraft to take off successfully. The diagram below shows an object \(A\), of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box \(B\) by a light inextensible string which passes over a smooth pulley \(P\), fixed at the edge of the surface, so that box \(B\) hangs freely. An object \(C\) lies on the horizontal floor of box \(B\) so that the combined mass of \(B\) and \(C\) is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to \(A\) in the direction \(P A\) so that box \(B\) is raised.
a) Find the magnitude of the acceleration of \(A\) and the tension in the string.
b) Given that object \(C\) has mass 4 kg , calculate the reaction of the floor of the box on object \(C\).
□
1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

• the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
• the park is 2 km away on a bearing of \(060 ^ { \circ }\).
  a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
  b) Determine the total distance walked by Sarah from her house to the park.
  c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 ms\(^{-2}\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.

1. Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
2. Find the car's average speed for the whole journey. [6 marks]

In reality the car's acceleration \(a\) ms\(^{-2}\) in the first 10 seconds is not constant, but increases from 0 to 4 ms\(^{-2}\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k(mt - t^2)\) for \(0 \leq t \leq 10\).

1. Calculate the values of the constants \(k\) and \(m\). [5 marks]
2. Find the acceleration of the car when \(t = 2\) according to this model. [2 marks]

A cyclist starting from rest accelerates uniformly at \(1.5 \text{ m s}^{-2}\) for \(4\) s and then travels at constant speed.

1. Sketch a velocity-time graph to represent the first \(10\) seconds of the cyclist's motion. [2]
2. Calculate the distance travelled by the cyclist in the first \(10\) seconds. [2]

5 A child is running up and down a path. A simplified model of the child's motion is as follows:

• he first runs north for 5 s at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\);
• he then suddenly stops and waits for 8 s ;
• finally he runs in the opposite direction for 7 s at \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Taking north to be the positive direction, sketch a velocity-time graph for this model of the child's motion.

Using this model,

calculate the total distance travelled by the child,

find his final displacement from his original position.

A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).

1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
2. Find the deceleration of the car for \(0 < t < 8\). [2]
3. Find the value of \(T\). [4]

A particle moves along the \(x\)-axis with velocity, \(v\) ms\(^{-1}\), at time \(t\) given by $$v = 24t - 6t^2.$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\). [2]
2. Find the times, \(t_1\) and \(t_2\), at which the particle has zero speed. [2]
3. Find the distance travelled between the times \(t_1\) and \(t_2\). [4]

A particle \(P\) moves in a straight line. The velocity \(v \text{ ms}^{-1}\) at time \(t\) s is given by $$v = 2t + 1 \quad \text{for } 0 \leqslant t \leqslant 5,$$ $$v = 36 - t^2 \quad \text{for } 5 \leqslant t \leqslant 7,$$ $$v = 2t - 27 \quad \text{for } 7 \leqslant t \leqslant 13.5.$$

1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\). [3]
2. Find the acceleration at the instant when \(t = 6\). [2]
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\). [5]

1. A man walks in a straight line from \(A\) to \(B\) with constant acceleration \(0.004 \text{ m s}^{-2}\). His speed at \(A\) is \(1.8 \text{ m s}^{-1}\) and his speed at \(B\) is \(2.2 \text{ m s}^{-1}\). Find the time taken for the man to walk from \(A\) to \(B\), and find the distance \(AB\). [3]
2. A woman cyclist leaves \(A\) at the same instant as the man. She starts from rest and travels in a straight line to \(B\), reaching \(B\) at the same instant as the man. At time \(t\) s after leaving \(A\) the cyclist's speed is \(k(200t - t^2) \text{ m s}^{-1}\), where \(k\) is a constant. Find
   1. the value of \(k\), [4]
   2. the cyclist's speed at \(B\). [1]
3. Sketch, using the same axes, the velocity-time graphs for the man's motion and the woman's motion from \(A\) to \(B\). [3]

An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s\(^{-1}\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.

1. In the space below, sketch a speed-time graph to illustrate the motion of the athlete. [3]
2. Calculate the value of \(T\). [5]

A tram starts from rest and moves with uniform acceleration for 20 s. The tram then travels at a constant speed, \(V \text{ ms}^{-1}\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km.

1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving. [2]
2. Find \(V\). [2]
3. Find the magnitude of the acceleration. [2]

A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).

1. Sketch the velocity-time graph for the jogger's journey. [2]
2. Show that \(3V^2 - 100V + 352 = 0\). [6]
3. Hence find the value of \(V\), giving a reason for your answer. [3]

The points \(P\) and \(Q\) are at the same height \(h\) metres above horizontal ground. A small stone is dropped from rest from \(P\). Half a second later a second small stone is thrown vertically downwards from \(Q\) with speed 7.35 m s\(^{-1}\). Given that the stones hit the ground at the same time, find the value of \(h\). [7]

A car moves in a straight line with initial speed \(u\) m s\(^{-1}\) and constant acceleration \(a\) m s\(^{-2}\). The car takes 5 s to travel the first 80 m and it takes 8 s to travel the first 160 m. Find \(a\) and \(u\). [6]

An object is moving in a straight line, with constant acceleration \(a\text{ m s}^{-2}\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below. \includegraphics{figure_13} Use the graph to show that $$v = u + at$$ [3 marks]

1. The model gives the correct velocity of \(25.6 \mathrm {~ms} ^ { - 1 }\) at time 8 s . Show that \(k = 0.1\). A second model for the motion uses constant acceleration.
2. Find the value of the acceleration which gives the correct velocity of \(25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time 8 s .
3. Show that these two models give the same value for the displacement in the first 8 s .

A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$

1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
2. Find the times when the particle is at rest. [2]

\includegraphics{figure_2} The diagram shows a wire \(ABCD\) consisting of a straight part \(AB\) of length \(5\) m and a part \(BCD\) in the shape of a semicircle of radius \(6\) m and centre \(O\). The diameter \(BD\) of the semicircle is horizontal and \(AB\) is vertical. A small ring is threaded onto the wire and slides along the wire. The ring starts from rest at \(A\). The part \(AB\) of the wire is rough, and the ring accelerates at a constant rate of \(2.5\) m s\(^{-2}\) between \(A\) and \(B\).

1. Show that the speed of the ring as it reaches \(B\) is \(5\) m s\(^{-1}\). [1]

A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
2. Find the time taken by the car to travel from \(P\) to \(S\). [3]

In taking off, an aircraft moves on a straight runway \(AB\) of length 1.2 km. The aircraft moves from \(A\) with initial speed \(2 \text{ m s}^{-1}\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \text{ m s}^{-1}\). Find

1. the acceleration of the aircraft, [2]
2. the distance \(BC\). [4]

A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find

1. the value of \(u\) and the acceleration of the particle, [7]
2. the value of \(\alpha\) in degrees. [2]

A body moves in a straight line with constant acceleration. Its speed increases from 17 ms\(^{-1}\) to 33 ms\(^{-1}\) while it travels a distance of 250 m. Find

1. the time taken to travel the 250 m, \hfill [3 marks]
2. the acceleration of the body. \hfill [2 marks]

The body now decelerates at a constant rate from 33 ms\(^{-1}\) to rest in 6 seconds.

1. Find the distance travelled in these 6 seconds. \hfill [2 marks]

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 \(A B\).

Two trucks \(P\) and \(Q\), of masses 18 000 kg and 16 000 kg respectively, collide while moving towards each other in a straight line. Immediately before the collision, both trucks are travelling at the same speed, \(u\) ms\(^{-1}\). Immediately after the collision, \(P\) is moving at half its original speed, its direction of motion having been reversed.

1. Find, in terms of \(u\), the speed of \(Q\) immediately after the collision. \hfill [5 marks]
2. State, with a reason, whether the direction of \(Q\)'s motion has been reversed. \hfill [1 mark]
3. Find, in terms of \(u\), the magnitude of the impulse exerted by \(P\) on \(Q\) in the collision, stating the units of your answer. \hfill [3 marks]

The force exerted by each truck on the other in the impact has magnitude \(108000u\) N.

1. Find the time for which the trucks are in contact. \hfill [3 marks]

5 A particle \(P\) moves in a straight line \(A B C D\) with constant deceleration. The velocities of \(P\) at \(A , B\) and \(C\) are \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Find the ratio of distances \(A B : B C\).
2. The particle comes to rest at \(D\). Given that the distance \(A D\) is 80 m , find the distance \(B C\).

6. A particle \(P\) is moving with constant acceleration. At time \(t = 1\) second, \(P\) has velocity \(( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = 4\) seconds, \(P\) has velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Find the speed of \(P\) at time \(t = 3.5\) seconds.