3.02h Motion under gravity: vector form

1. At time \(t = 0\), a small stone is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point \(A\).

At time \(t = T\) seconds, the stone passes through \(A\), moving downwards.
The stone is modelled as a particle moving freely under gravity throughout its motion.
Using the model,

1. find the value of \(T\),
2. find the total distance travelled by the stone in the first 4 seconds of its motion.
3. State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

5. \begin{figure}[h] \end{figure} A small ball is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The point \(O\) is 25 m vertically above the point \(N\) which is on horizontal ground. The ball is projected at an angle of \(45 ^ { \circ }\) above the horizontal.
The ball hits the ground at a point \(A\), where \(A N = 100 \mathrm {~m}\), as shown in Figure 2 .
The motion of the ball is modelled as that of a particle moving freely under gravity.
Using this initial model,

1. show that \(U = 28\)
2. find the greatest height of the ball above the horizontal ground \(N A\). In a refinement to the model of the motion of the ball from \(O\) to \(A\), the effect of air resistance is included. This refined model is used to find a new value of \(U\).
3. How would this new value of \(U\) compare with 28, the value given in part (a)?
4. State one further refinement to the model that would make the model more realistic. \section*{" " \(_ { \text {" } } ^ { \text {" } }\) " "}

4. \begin{figure}[h] \end{figure} A small stone is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
Point \(O\) is 70 m vertically above the point \(N\).
Point \(N\) is on horizontal ground.
The stone is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The stone hits the ground at the point \(A\), as shown in Figure 3.
The stone is modelled as a particle moving freely under gravity.
The acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 m ~ s } \mathbf { m ~ } ^ { \mathbf { - 2 } }\) Using the model,

1. find the time taken for the stone to travel from \(O\) to \(A\),
2. find the speed of the stone at the instant just before it hits the ground at \(A\). One limitation of the model is that it ignores air resistance.
3. State one other limitation of the model that could affect the reliability of your answers.

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

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 \(t\).

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.

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 \(d\).

9 A child throws a pebble of mass 40 g vertically downwards with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.8 m above a sandy beach.

1. Calculate the speed at which the pebble hits the beach. The pebble travels 3 cm through the sand before coming to rest.
2. Find the magnitude of the resistance force of the sand on the pebble, assuming it is constant. Give your answer correct to \(\mathbf { 3 }\) significant figures.

14 A man runs at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. A woman is standing on a bridge that spans the road. At the instant that the man passes directly below the woman she throws a ball with initial speed \(u \mathrm {~ms} ^ { - 1 }\) at \(\alpha ^ { \circ }\) above the horizontal. The path of the ball is directly above the road. The man catches the ball 2.4 s after it is thrown. At the instant the man catches it, the ball is 3.6 m below the level of the point of projection.

1. Explain what it means that the ball is modelled as a particle.
2. Find the vertical component of the ball's initial velocity.
3. Find each of the following.

13 A projectile is fired from ground level at \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal.

1. State a modelling assumption that is used in the standard projectile model.
2. Find the cartesian equation of the trajectory of the projectile. The projectile travels above horizontal ground towards a wall that is 110 m away from the point of projection and 5 m high. The projectile reaches a maximum height of 22.5 m .
3. Determine whether the projectile hits the wall.

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 \(\alpha\).

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\). \(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).

1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}

7 The training rig for a parachutist comprises a fixed platform and a fixed hook, \(H\). The platform is 3.5 m above horizontal ground level. The hook, which is not directly above the platform, is 6.5 m above the ground. One end of a light inextensible cord of length 4.5 m is attached to \(H\) and the other is attached to a trainee parachutist of mass 90 kg standing on the edge of the platform with the cord straight and taut. The trainee is then projected off the platform with a velocity of \(7 \mathrm {~ms} ^ { - 1 }\) perpendicular to the cord in a downward direction. The motion of the trainee all takes place in a single vertical plane and while the cord is attached to \(H\) it remains straight and taut. When the speed of the trainee reaches \(5.5 \mathrm {~ms} ^ { - 1 }\) the cord is detached from \(H\) and the trainee then moves under the influence of gravity alone until landing on the ground (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-6_615_1211_934_242} The trainee is modelled as a particle and air resistance is modelled as being negligible.

1. Show that at the instant before the cord is detached from \(H\), the tension in the cord has a magnitude of 1005.5 N . The point on the ground vertically below the edge of the platform is denoted by \(O\). The point on the ground where the trainee lands is denoted by \(T\).
2. Determine the distance \(O T\). The ground around \(T\) is in fact an elastic mat of thickness 0.5 m which is angled so that it is perpendicular to the direction of motion of the trainee on landing. The mat, which is very rough, is modelled as an elastic spring of natural length 0.5 m . It is assumed that the trainee strikes the mat at ground level and is brought to rest once the mat has been compressed by 0.3 m .
3. Determine the modulus of elasticity of the mat. Give your answer to the nearest integer.

4 A light inextensible taut rope, of length 4 m , is attached at one end \(A\) to the centre of the horizontal ceiling of a gym. The other end of the rope \(B\) is being held by a child of mass 35 kg . Initially the child is held at rest with the rope making an angle of \(60 ^ { \circ }\) to the downward vertical and it may be assumed that the child can be modelled as a particle attached to the end of the rope. The child is released at a height 5 m above the horizontal ground.

1. Show that the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the child when the rope makes an angle \(\theta\) with the downward vertical is given by \(v ^ { 2 } = 4 g ( 2 \cos \theta - 1 )\).
2. At the instant when \(\theta = 0 ^ { \circ }\), the child lets go of the rope and moves freely under the influence of gravity only. Determine the speed and direction of the child at the moment that the child reaches the ground.
3. The child returns to the initial position and is released again from rest. Find the value of \(\theta\) when the tension in the rope is three times greater than the tension in the rope at the instant the child is released.

8 A football is placed on a horizontal surface. It is then kicked, so that it has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal.

1. State two modelling assumptions that it would be appropriate to make when considering the motion of the football.
2. 1. Find the time that it takes for the ball to reach its maximum height.
   2. Hence show that the maximum height of the ball is 3.04 metres, correct to three significant figures.
3. After the ball has reached its maximum height, it hits the bar of a goal at a height of 2.44 metres. Find the horizontal distance of the goal from the point where the ball was kicked.

7 A golf ball is struck from a point on horizontal ground so that it has an initial velocity of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. Assume that the golf ball is a particle and its weight is the only force that acts on it once it is moving.

1. Find the maximum height of the golf ball.
2. After it has reached its maximum height, the golf ball descends but hits a tree at a point which is at a height of 6 metres above ground level. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-5_289_1358_813_335} \begin{displayquote} Find the time that it takes for the ball to travel from the point where it was struck to the tree. \end{displayquote}

7 A golfer hits a ball which is on horizontal ground. The ball initially moves with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. There is a pond further along the horizontal ground. The diagram below shows the initial position of the ball and the position of the pond. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-5_387_1230_502_395}

1. State two assumptions that you should make in order to model the motion of the ball.
   (2 marks)
2. Show that the horizontal distance, in metres, travelled by the ball when it returns to ground level is $$\frac { V ^ { 2 } \sin 40 ^ { \circ } \cos 40 ^ { \circ } } { 4.9 }$$
3. Find the range of values of \(V\) for which the ball lands in the pond.

8 A cricket ball is hit at ground level on a horizontal surface. It initially moves at \(28 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal.

1. Find the maximum height of the ball during its flight.
2. The ball is caught when it is at a height of 2 metres above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-5_332_1070_1601_477} Show that the time that it takes for the ball to travel from the point where it was hit to the point where it was caught is 4.28 seconds, correct to three significant figures.
3. Find the speed of the ball when it is caught.

7 An arrow is fired from a point at a height of 1.5 metres above horizontal ground. It has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. The arrow hits a target at a height of 1 metre above horizontal ground. The path of the arrow is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-18_341_1260_550_390} Model the arrow as a particle.

1. Show that the time taken for the arrow to travel to the target is 1.30 seconds, correct to three significant figures.
2. Find the horizontal distance between the point where the arrow is fired and the target.
3. Find the speed of the arrow when it hits the target.
4. Find the angle between the velocity of the arrow and the horizontal when the arrow hits the target.
5. State one assumption that you have made about the forces acting on the arrow.
   (1 mark)
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-19_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-20_2486_1714_221_153}

8 A girl stands at the edge of a quay and sees a tin can floating in the water. The water level is 5 metres below the top of the quay and the can is at a horizontal distance of 10 metres from the quay, as shown in the diagram.
The girl decides to throw a stone at the can. She throws the stone from a height of 1 metre above the top of the quay. The initial velocity of the stone is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) below the horizontal, so that the initial velocity of the stone is directed at the can, as shown in the diagram.
Assume that the stone is a particle and that it experiences no air resistance as it moves.

1. Find \(\alpha\).
2. Find the time that it takes for the stone to reach the level of the water.
3. Find the distance between the stone and the can when the stone hits the water.

6 A ball is kicked from the point \(P\) on a horizontal surface. It leaves the surface with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal and hits the surface for the first time at the point \(Q\). Assume that the ball is a particle that moves only under the influence of gravity. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-12_317_1118_513_461}

1. Show that the time that it takes the ball to travel from \(P\) to \(Q\) is 3.13 s , correct to three significant figures.
2. Find the distance between the points \(P\) and \(Q\).
3. If a heavier ball were projected from \(P\) with the same velocity, how would the distance between \(P\) and \(Q\), calculated using the same modelling assumptions, compare with your answer to part (b)? Give a reason for your answer.
4. Find the maximum height of the ball above the horizontal surface.
5. State the magnitude and direction of the velocity of the ball as it hits the surface.
   
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-13_2484_1709_223_153}

6 Two particles, \(A\) and \(B\), have masses 12 kg and 8 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg, as shown in the diagram. $$A ( 12 \mathrm {~kg} )$$ The particles are released from rest and move vertically. Assume that there is no air resistance.

1. By forming two equations of motion, show that the magnitude of the acceleration of each particle is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. After the particles have been moving for 2 seconds, both particles are at a height of 4 metres above a horizontal surface. When the particles are in this position, the string breaks.
   1. Find the speed of particle \(A\) when the string breaks.
   2. Find the speed of particle \(A\) when it hits the surface.
   3. Find the time that it takes for particle \(B\) to reach the surface after the string breaks. Assume that particle \(B\) does not hit the peg.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-13_2484_1709_223_153}

8 A ball is struck so that it leaves a horizontal surface travelling at \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-16_293_1364_461_347}

1. Show that the ball takes \(\frac { 3 \sin \alpha } { 2 }\) seconds to reach its maximum height.
2. The ball reaches a maximum height of 7 metres.
   1. Find \(\alpha\).
   2. Find the range, \(O A\).
3. State two assumptions that you needed to make in order to answer the earlier parts of this question. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-17_2347_1691_223_153} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-18_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-19_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-20_2505_1734_212_138}