3.02d Constant acceleration: SUVAT formulae

7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
Ball \(A\) is modelled as a particle moving freely under gravity.

1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
   Given that the mass of \(A\) is 0.1 kg ,
2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
   At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
3. Find the value of \(T _ { 2 }\)

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 \(T\).

11 A particle \(P\) of mass 2 kg can move along a line of greatest slope on the smooth surface of a wedge which is fixed to the ground. The sloping face \(O A\) of the wedge has length 10 metres and is inclined at \(30 ^ { \circ }\) to the horizontal (see Fig. 1). \(P\) is fired up the slope from the lowest point \(O\), with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Find the time taken for \(P\) to reach \(A\) and show that the speed of \(P\) at \(A\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) has reached \(A\) it becomes a projectile (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_424_1533_1123_306} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. Find the total horizontal distance travelled by \(P\) from \(O\) when it hits the ground.

8 A tram travels from stop \(A\) to stop \(B\), a distance of 300 m . First the tram starts from rest at \(A\) and accelerates uniformly at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 16 seconds. Then it travels at a constant speed and finally it slows down uniformly at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) coming to rest at \(B\).

1. Sketch the velocity-time graph for the journey of the tram from \(A\) to \(B\).
2. Find the speed of the tram and the distance travelled at the end of the first 16 seconds.
3. Show that the journey from \(A\) to \(B\) takes 49.5 seconds.

10 A particle is projected up a long smooth slope at a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The slope is at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 25 }\). After 2 seconds it passes a mark on the slope. Find the total time taken from the moment of projection until it passes the mark again and the total distance travelled in that time. {www.cie.org.uk} after the live examination series. }

6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.

1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
   Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
2. Find the tension in the cable in each of the three stages.
3. Sketch the velocity-time graph for the complete downward motion of the crate.
4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.

9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.

1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
   1. Write down the values of \(t\) when \(v = 0\).
   2. Show that \(x = 5.6\) when \(t = 18\).
   3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.

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. }

8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find

1. the greatest height above the ground reached by the ball,
2. the time taken for the ball to reach the ground.

12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.

1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.

12

1. Whilst a helicopter is hovering, the floor of its cargo hold maintains an angle of \(30 ^ { \circ }\) to the horizontal. There is a box of mass 20 kg on the floor. If the box is just on the point of sliding, show by resolving forces that the coefficient of friction between the box and the floor is \(\frac { 1 } { \sqrt { 3 } }\).
2. The helicopter ascends at a constant acceleration 0.5 g . If the cargo hold is now maintained at \(10 ^ { \circ }\) to the horizontal, determine the frictional force and the normal reaction between the box and the floor.

13 Professor Oldham wishes to illustrate and test Newton's experimental law of impacts. A ball is dropped from rest from a height \(h\) above a rigid horizontal board and rebounds to a height \(H\). The time taken to reach the height \(H\) after the first impact is \(T\). These quantities are recorded using very accurate measuring devices.

1. Show that $$H = e ^ { 2 } h \quad \text { and } \quad T = e \sqrt { \frac { 2 h } { g } }$$ are predicted by Newton's law, where \(e\) is the coefficient of restitution between the ball and the board.
2. If \(h = 180 \mathrm {~cm}\) and \(H = 45 \mathrm {~cm}\), determine \(T\) from these formulae. The experiment is repeated for initial heights \(h , 2 h , 3 h , \ldots , 15 h\) where \(h = 180 \mathrm {~cm}\). The corresponding rebound heights and times taken to reach that height after the first impact are recorded. The mean of the 15 rebound heights is found to be 3.3 m .
3. Find the mean of the rebound heights predicted by Newton's law and give one reason why this differs from the experimental value. Professor Oldham is able to repeat the experiment on the surface of the moon using the same experimental set-up inside a laboratory.
4. The mean of the rebound heights is unchanged, but the mean of the rebound times is substantially increased. Comment on these findings.

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 \(C\).
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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 particle \(P\) is projected vertically upwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) which is \(2.8\text{ m}\) above horizontal ground.

1. Find the greatest height above the ground reached by \(P\). [3]
2. Find the length of time for which \(P\) is at a height of more than \(3.6\text{ m}\) above the ground. [4]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [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]

Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point \(A\). Isabella accelerates for 5 s at a constant rate \(a \text{ m s}^{-2}\). She then travels at the constant speed she has reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s. Maria accelerates at a constant rate, reaching a speed of \(5 \text{ m s}^{-1}\) in a distance of 27.5 m. She then maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period of 5 s.

1. Given that \(a = 1.1\), find which cyclist travels further. [5]
2. Find the value of \(a\) for which the two cyclists travel the same distance. [2]

A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m, reaching a speed of 25 m s\(^{-1}\). The car then travels at this speed for 400 m, before decelerating uniformly to rest over a period of 5 s.

1. Find the time for which the car is accelerating. [2]
2. Sketch the velocity–time graph for the motion of the car, showing the key points. [2]
3. Find the average speed of the car during its motion. [2]

A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$

A particle is projected vertically upwards from horizontal ground. The speed of the particle 2 seconds after it is projected is \(5\) m s\(^{-1}\) and it is travelling downwards.

1. Find the speed of projection of the particle. [2]
2. Find the distance travelled by the particle between the two times at which its speed is \(10\) m s\(^{-1}\). [2]

\includegraphics{figure_7} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(ABC\). Particles \(P\) and \(Q\) are each of mass \(m\) kg. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(AB\) is 0.75 m and the length of \(BC\) is 3.25 m. The section \(AB\) of the plane is smooth and the section \(BC\) is rough. The coefficient of friction between each particle and the section \(BC\) is 0.25. Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).

1. Verify that particle \(P\) reaches \(B\) 0.5 s after it is released, with speed \(3\) m s\(^{-1}\). [3]
2. Find the time that it takes from the instant the two particles are released until they collide. [4]

The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25.

1. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\). [5]

\includegraphics{figure_4} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light string that makes an angle of 26° above a line of greatest slope. The tension in the string is \(T\) N (see diagram). The coefficient of friction between the block and plane is 0.65.

1. The acceleration of the block is 0.2 m s\(^{-2}\). Find \(T\). [7]
2. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion. [2]

\includegraphics{figure_5} A block \(A\) of mass 80 kg is connected by a light, inextensible rope to a block \(B\) of mass 40 kg. The rope joining the two blocks is taut and is parallel to a line of greatest slope of a plane which is inclined at an angle of \(20°\) to the horizontal. A force of magnitude 500 N inclined at an angle of \(15°\) above the same line of greatest slope acts on \(A\) (see diagram). The blocks move up the plane and there is a resistance force of 50 N on \(B\), but no resistance force on \(A\).

1. Find the acceleration of the blocks and the tension in the rope. [5]

1. Find the time that it takes for the blocks to reach a speed of \(1.2 \text{ m s}^{-1}\) from rest. [2]

Three particles \(A\), \(B\) and \(C\) of masses 0.3 kg, 0.4 kg and \(m\) kg respectively lie at rest in a straight line on a smooth horizontal plane. The distance between \(B\) and \(C\) is 2.1 m. \(A\) is projected directly towards \(B\) with speed \(2 \text{ m s}^{-1}\). After \(A\) collides with \(B\) the speed of \(A\) is reduced to \(0.6 \text{ m s}^{-1}\), still moving in the same direction.

1. Show that the speed of \(B\) after the collision is \(1.05 \text{ m s}^{-1}\). [2]

After the collision between \(A\) and \(B\), \(B\) moves directly towards \(C\). Particle \(B\) now collides with \(C\). After this collision, the two particles coalesce and have a combined speed of \(0.5 \text{ m s}^{-1}\).

1. Find \(m\). [2]

1. Find the time that it takes, from the instant when \(B\) and \(C\) collide, until \(A\) collides with the combined particle. [5]

A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).

1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane. [3]
2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3, find the distance travelled by \(P\) in the third second of its motion. [4]