Constant acceleration (SUVAT)

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.

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 \(t\).

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 block of mass 3 kg is initially at rest on a smooth horizontal floor. A force of 12 N , acting at an angle of \(25 ^ { \circ }\) above the horizontal, is applied to the block. Find the distance travelled by the block in the first 5 seconds of its motion.

1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$

1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the total distance travelled by the particle during the 15 seconds.

4 A cyclist travels along a straight road with constant acceleration. He passes through points \(A , B\) and \(C\). The cyclist takes 2 seconds to travel along each of the sections \(A B\) and \(B C\) and passes through \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(A B\) is \(\frac { 4 } { 5 }\) of the distance \(B C\).

1. Find the acceleration of the cyclist.
2. Find \(A C\).

2 The graph shows how the speed of a cyclist, Hannah, varies as she travels for 21 seconds along a straight horizontal road. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-2_590_1603_847_230}

1. Find the distance travelled by Hannah in the 21 seconds.
2. Find Hannah's average speed during the 21 seconds.

A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity–time graph below: \includegraphics{figure_13} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. [1 mark] Tick \((\checkmark)\) one box. The car never accelerates. The acceleration of the car is always positive. The acceleration of the car can change instantaneously. The acceleration of the car is never constant.

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.

1 A lift moves upwards from rest and accelerates at \(0.9 \mathrm {~ms} ^ { - 2 }\) for 3 s . The lift then travels for 6 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 4 s .

1. Sketch a velocity-time graph for the motion.
2. Find the total distance travelled by the lift.

1 A lift moves upwards from rest and accelerates at \(0.9 \mathrm {~ms} ^ { - 2 }\) for 3 s . The lift then travels for 6 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 4 s .

1. Sketch a velocity-time graph for the motion.
2. Find the total distance travelled by the lift.

7. Two cyclists, Alice and Bobbie, travel from \(P\) to \(Q\) along a straight path. Alice starts from rest at \(P\) just as Bobbie passes her at \(3.5 \mathrm {~ms} ^ { - 1 }\). Bobbie continues at this speed while Alice accelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds until she attains her maximum speed. At this moment both cyclists immediately start to slow down, with constant but different decelerations, and they come to rest at \(Q 80\) seconds after Alice started moving.

1. Sketch, on the same diagram, the velocity-time graphs for the two cyclists. By using the fact that both cyclists cover the same distance, find
2. the value of \(T\),
3. the distance between \(P\) and \(Q\),
4. the magnitude of Bobbie's deceleration.

1 An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?

In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground. Calculate the speed of the ball when it reaches the ground. Circle your answer. [1 mark] \(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)

1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.

1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and north respectively.]

1. A particle \(P\) is moving with constant acceleration ( \(- 4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { ms } ^ { - 2 }\)

At time \(t = 0 , P\) has velocity \(( 14 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the speed of \(P\) at time \(t = 2\) seconds.
2. Find the size of the angle between the direction of \(\mathbf { i }\) and the direction of motion of \(P\) at time \(t = 2\) seconds. At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(2 \mathbf { i } - 3 \mathbf { j }\) )
3. Find the value of \(T\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.]

A particle \(P\) is moving with constant acceleration. At 2 pm , the velocity of \(P\) is \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and at 2.30 pm the velocity of \(P\) is \(( \mathbf { i } + 7 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At time \(T\) hours after \(2 \mathrm { pm } , P\) is moving in the direction of the vector \(( - \mathbf { i } + 2 \mathbf { j } )\)

1. Find the value of \(T\). Another particle, \(Q\), has velocity \(\mathbf { v } _ { Q } \mathrm {~km} \mathrm {~h} ^ { - 1 }\) at time \(t\) hours after 2 pm , where $$\mathbf { v } _ { Q } = ( - 4 - 2 t ) \mathbf { i } + ( \mu + 3 t ) \mathbf { j }$$ and \(\mu\) is a constant. Given that there is an instant when the velocity of \(P\) is equal to the velocity of \(Q\),
2. find the value of \(\mu\).

1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
2. Find \(T\).
3. Calculate the distance \(A\) and \(B\) each travel.

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.

6 Particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. \(P\) is released from rest at a point \(O\) on the line and 2 s later passes through the point \(A\) with speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) and the angle of inclination of the plane. At the instant that \(P\) passes through \(A\) the particle \(Q\) is released from rest at \(O\). At time \(t\) s after \(Q\) is released from \(O\), the particles \(P\) and \(Q\) are 4.9 m apart.
2. Find the value of \(t\).

1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).

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
   VJYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JLIYM ION OC

7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). 24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
2. Find the time that it takes from when the cyclist starts until the car overtakes her.

5 A car of mass 1200 kg travels from rest along a straight horizontal road. The driving force is 4000 N and the total of all resistances to motion is 800 N .
Calculate the velocity of the car after 9 seconds.

1. A car is initially at rest on a straight horizontal road.

The car then accelerates along the road with a constant acceleration of \(3.2 \mathrm {~ms} ^ { - 2 }\) Find

1. the speed of the car after 5 s ,
2. the distance travelled by the car in the first 5 s .

7 A walker travels along a straight road passing through the points \(A\) and \(B\) on the road with speeds \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The walker's acceleration between \(A\) and \(B\) is constant and equal to \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time taken by the walker to travel from \(A\) to \(B\), and find the distance \(A B\). A cyclist leaves \(A\) at the same instant as the walker. She starts from rest and travels along the straight road, passing through \(B\) at the same instant as the walker. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant.
2. Show that when \(t = 64.05\) the speed of the walker and the speed of the cyclist are the same, correct to 3 significant figures.
3. Find the cyclist's acceleration at the instant she passes through \(B\).

8 A crane lifts a crate of mass 20 kg using a light inextensible cable. The crate starts from rest and ascends 10 metres in 4 seconds during which time a constant tension of \(T \mathrm {~N}\) is applied in the cable. Find the value of \(T\).

1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the particle,
2. the value of \(\alpha\).

4 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438} \(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find

1. the values of \(u\) and \(a\),
2. the value of \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.

1. 10 seconds after passing a warning signal, a train is travelling at \(18 m s ^ { - 1 }\) and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal.

\section*{BLANK PAGE FOR WORKING}

2 A lift is travelling upwards and accelerating uniformly. During a 5 second period, it travels 16 metres and the speed of the lift increases from \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. \(\quad\) Find \(u\).
2. Find the acceleration of the lift.

4 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h] \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).

1. Find the initial velocity and the acceleration of the particle.
2. Prove that the particle is never at P .

3. A lorry accelerates uniformly from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds.

1. Find how far it travels while accelerating.
2. Find, in seconds correct to 2 decimal places, the length of time it takes for the lorry to cover the first half of this distance.
   (6 marks)

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.

7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.

\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.

1. Calculate the deceleration of car \(A\). [2]
2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]

4 A particle \(P\) travels in the positive direction along a straight line with constant acceleration. \(P\) travels a distance of 52 m during the 2 nd second of its motion and a distance of 64 m during the 4th second of its motion.

1. Find the initial speed and the acceleration of \(P\).
2. Find the distance travelled by \(P\) during the first 10 seconds of its motion.

5. A cyclist, Laura, is travelling in a straight line on a horizontal road at a constant speed of \(25 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) A second cyclist, Jason, is riding closely and directly behind Laura. He is also moving with a constant speed of \(25 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

1. The driving force applied by Jason is likely to be less than the driving force applied by Laura. Explain why.
2. Jason has a problem and stops, but Laura continues at the same constant speed. Laura sees an accident 40 m ahead, so she stops pedalling and applies the brakes.
   She experiences a total resistance force of 40 N
   Laura and her cycle have a combined mass of 64 kg
   (b) (i) Determine whether Laura stops before reaching the accident. Fully justify your answer.
   [0pt] [4 marks]
   (b) (ii) State one assumption you have made that could affect your answer to part (b)(i).

6 A particle of mass 3 kg falls from rest at a point 5 m above the surface of a liquid which is in a container. There is no instantaneous change in speed of the particle as it enters the liquid. The depth of the liquid in the container is 4 m . The downward acceleration of the particle while it is moving in the liquid is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the resistance to motion of the particle while it is moving in the liquid.
2. Sketch the velocity-time graph for the motion of the particle, from the time it starts to move until the time it reaches the bottom of the container. Show on your sketch the velocity and the time when the particle enters the liquid, and when the particle reaches the bottom of the container.