3.02d Constant acceleration: SUVAT formulae

11 \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).

1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
   (b) Find the acceleration of \(P\). \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
   It is given that
   \section*{OCR} \section*{Oxford Cambridge and RSA}

10 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.

1. Determine the values of \(p , q\) and \(r\).
2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).

11 Two small balls \(P\) and \(Q\) have masses 3 kg and 2 kg respectively. The balls are attached to the ends of a string. \(P\) is held at rest on a rough horizontal surface. The string passes over a pulley which is fixed at the edge of the surface. \(Q\) hangs vertically below the pulley at a height of 2 m above a horizontal floor. \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-7_346_906_445_255} The system is initially at rest with the string taut. A horizontal force of magnitude 40 N acts on \(P\) as shown in the diagram. \(P\) is released and moves directly away from the pulley. A constant frictional force of magnitude 8 N opposes the motion of \(P\). It is given that \(P\) does not leave the horizontal surface and that \(Q\) does not reach the pulley in the subsequent motion. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the string is modelled as being light and inextensible.

1. Show that the magnitude of the acceleration of each particle is \(2.48 \mathrm {~ms} ^ { - 2 }\).
2. Find the tension in the string. When the balls have been in motion for 0.5 seconds, the string breaks.
3. Find the additional time that elapses until \(Q\) hits the floor.
4. Find the speed of \(Q\) as it hits the floor.
5. Write down the magnitude of the normal reaction force acting on \(Q\) when \(Q\) has come to rest on the floor.
6. State one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA

11 A car starts from rest at a set of traffic lights and moves along a straight road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A motorcycle, travelling parallel to the car with constant speed \(16 \mathrm {~ms} ^ { - 1 }\), passes the same traffic lights exactly 1.5 seconds after the car starts to move. The time after the car starts to move is denoted by \(t\) seconds.

1. Determine the two values of \(t\) at which the car and motorcycle are the same distance from the traffic lights. These two values of \(t\) are denoted by \(t _ { 1 }\) and \(t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Describe the relative positions of the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\).
3. Determine the maximum distance between the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\). \section*{END OF QUESTION PAPER}

10 A cyclist starts from rest and moves with constant acceleration along a straight horizontal road. The cyclist reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) in 25 seconds. The cyclist then moves with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until the speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist then moves with constant deceleration until coming to rest. The total time for the cyclist's journey is 150 seconds.

1. Sketch a velocity-time graph to represent the cyclist's motion.
2. Find the acceleration during the first 25 seconds of the cyclist's motion. The cyclist takes \(T\) seconds to decelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until coming to rest.
3. Determine the value of \(T\).
4. Determine the average speed for the cyclist's journey.

11 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-08_451_1340_251_244} A train consists of an engine \(A\) of mass 50000 kg and a carriage \(B\) of mass 20000 kg . The engine and carriage are connected by a rigid coupling. The coupling is modelled as light and horizontal. The resistances to motion acting on \(A\) and \(B\) are 9000 N and 1250 N respectively (see diagram).
The train passes through station \(P\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along a straight horizontal track with constant acceleration \(0.01 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) towards station \(Q\). The distance between \(P\) and \(Q\) is 12.95 km .

1. Determine the time, in minutes, to travel between \(P\) and \(Q\). For the train's motion between \(P\) and \(Q\), determine the following.
2. The driving force of the engine.
3. The tension in the coupling between \(A\) and \(B\).

10 A small ball \(B\) is projected vertically upwards from a point 2 m above horizontal ground. \(B\) is projected with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\), and takes \(t\) seconds to reach the ground. Find the value of \(t\).

9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Find the time, in seconds, for which the cyclist is decelerating.
2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
3. Find the time, in seconds, for which the cyclist is moving at constant velocity.

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.

10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.

1. Sketch the velocity-time graph for Rory's run using this model.
2. Calculate \(T\).
3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
5. Sketch a velocity-time graph that represents Rory's run using this refined model.
6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)

11 A sports car accelerates along a straight road from rest. After 5 s its velocity is \(9 \mathrm {~ms} ^ { - 1 }\). In model A, the acceleration is assumed to be constant.

1. Calculate the distance travelled by the car in the first 5 seconds according to model A . In model B , the velocity \(v\) in \(\mathrm { ms } ^ { - 1 }\) is given by \(\mathrm { v } = 0.05 \mathrm { t } ^ { 3 } + \mathrm { kt }\), where \(t\) is the time in seconds after the start and \(k\) is a constant.
2. Find the value of \(k\) which gives the correct value of \(v\) when \(t = 5\).
3. Using this value of \(k\) in model B , calculate the acceleration of the car when \(t = 5\). The car travels 16 m in the first 5 seconds.
4. Show that model B, with the value of \(k\) found in part (b), better fits this information than model A does.

12 Points A, B and C lie in a straight line in that order on horizontal ground. A box of mass 5 kg is pushed from A to C by a horizontal force of magnitude 8 N . The box is at rest at A and takes 3 seconds to reach B . The ground is smooth between A and B . Between B and C the ground is rough and the resistance to motion is 28 N . The box comes to rest at C . Determine the distance AC.

9 A car travelling in a straight line accelerates uniformly from rest to \(V \mathrm {~ms} ^ { - 1 }\) in \(T \mathrm {~s}\). It then slows down uniformly, coming to rest after a further \(2 T\) s.

1. Sketch a velocity-time graph for the car. The acceleration in the first stage of the motion is \(2.5 \mathrm {~ms} ^ { - 2 }\) and the total distance travelled is 240 m .
2. Calculate the values of \(V\) and \(T\).

10 An astronaut on the surface of the moon drops a ball from a point 2 m above the surface.

1. Without any calculations, explain why a standard model using \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) will not be appropriate to model the fall of the ball. The ball takes 1.6s to hit the surface.
2. Find the acceleration of the ball which best models its motion. Give your answer correct to 2 significant figures.
3. Use this value to predict the maximum height of the ball above the point of projection when thrown vertically upwards with an initial velocity of \(15 \mathrm {~ms} ^ { - 1 }\).

2 An unmanned craft lands on the planet Mars. A small bolt falls from the craft onto the surface of the planet. It falls 1.5 m from rest in 0.9 s . Calculate the acceleration due to gravity on Mars.

4 Rory pushes a box of mass 2.8 kg across a rough horizontal floor against a resistance of 19 N . Rory applies a constant horizontal force. The box accelerates from rest to \(1.2 \mathrm {~ms} ^ { - 1 }\) as it travels 1.8 m .

1. Calculate the acceleration of the box.
2. Find the magnitude of the force that Rory applies.

9 A pebble is thrown horizontally at \(14 \mathrm {~ms} ^ { - 1 }\) from a window which is 5 m above horizontal ground. The pebble goes over a fence 2 m high \(d \mathrm {~m}\) away from the window as shown in Fig. 9. The origin is on the ground directly below the window with the \(x\)-axis horizontal in the direction in which the pebble is thrown and the \(y\)-axis vertically upwards. \begin{figure}[h] \end{figure}

1. Find the time the pebble takes to reach the ground.
2. Find the cartesian equation of the trajectory of the pebble.
3. Find the range of possible values for \(d\).

14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds. In model A, the acceleration is assumed to be uniform.

1. Find an expression for the velocity of the car at time \(t\) using this model.
2. Explain why this model is not appropriate in the long term. Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
4. Calculate the displacement of the car in the first 20 seconds according to this refined model. In model B, the velocity of the car is given by $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8 \\ 17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
6. Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
7. Show that model B gives the same value as model A for the displacement at time 20 s .

16 A particle of mass 2 kg slides down a plane inclined at \(20 ^ { \circ }\) to the horizontal. The particle has an initial velocity of \(1.4 \mathrm {~ms} ^ { - 1 }\) down the plane. Two models for the particle's motion are proposed. In model A the plane is taken to be smooth.

1. Calculate the time that model A predicts for the particle to slide the first 0.7 m .
2. Explain why model A is likely to underestimate the time taken. In model B the plane is taken to be rough, with a constant coefficient of friction between the particle and the plane.
3. Calculate the acceleration of the particle predicted by model B given that it takes 0.8 s to slide the first 0.7 m .
4. Find the coefficient of friction predicted by model B , giving your answer correct to 3 significant figures. \section*{END OF QUESTION PAPER}

1 A ball is thrown vertically upwards with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
Find the times at which the ball is 3 m above the point of projection.

8 A bus is travelling along a straight road at \(5.4 \mathrm {~ms} ^ { - 1 }\). At \(t = 0\), as the bus passes a boy standing on the pavement, the boy starts running in the same direction as the bus, accelerating at \(1.2 \mathrm {~ms} ^ { - 2 }\) from rest for 5 s . He then runs at constant speed until he catches up with the bus.

1. The diagram in the Printed Answer Booklet shows the velocity-time graph for the bus. Draw the velocity-time graph for the boy on this diagram.
2. Determine the time at which the boy is running at the same speed as the bus.
3. Find the maximum distance between the bus and the boy.
4. Find the distance the boy has run when he catches up with the bus.

11 A block of mass 2 kg is placed on a rough horizontal table. A light inextensible string attached to the block passes over a smooth pulley attached to the edge of the table. The other end of the string is attached to a sphere of mass 0.8 kg which hangs freely. The part of the string between the block and the pulley is horizontal. The coefficient of friction between the table and the block is 0.35 . The system is released from rest.

1. Draw a force diagram showing all the forces on the block and the sphere.
2. Write down the equations of motion for the block and the sphere.
3. Show that the acceleration of the system is \(0.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the time for the block to slide the first 0.5 m . Assume the block does not reach the pulley.

10 A ball is thrown upwards with a velocity of \(29.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the ball reaches its maximum height after 3 s .
2. Sketch a velocity-time graph for the first 5 s of motion.
3. Calculate the speed of the ball 5 s after it is thrown. A second ball is thrown at \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha ^ { \circ }\) above the horizontal. It reaches the same maximum height as the first ball.
4. Use this information to write down
   This second ball reaches its greatest height at a point which is 48 m horizontally from the point of projection.
5. Calculate the values of \(u\) and \(\alpha\).

12 A box of mass \(m \mathrm {~kg}\) slides down a rough slope inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between the box and the slope is 0.4 . The box has an initial velocity of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slope. Calculate the distance the box travels before coming to rest.

7 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A small stone is projected from a point O with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7.
You are given that \(\sin \theta = \frac { 12 } { 13 }\).
After \(t\) seconds the horizontal displacement of the stone from O is \(x\) metres and the vertical displacement is \(y\) metres. \begin{figure}[h] \end{figure}

1. Using the standard model for projectile motion,
   The stone passes through a point A . Point A is 16 m above the level of O .
2. Find the two possible horizontal distances of A from O . A toy balloon is projected from O with the same initial velocity as the small stone.
3. Suggest two ways in which the standard model could be adapted.