Basic trajectory calculations

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}

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}

2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h] \end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.

1. Find its initial speed.
2. Find the ball's flight time and range, \(R \mathrm {~m}\).
3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
   (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

2 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25 ms 1 and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).

1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
2. Calculate the maximum height reached by the ball.
3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
4. Determine the following when \(t = 1.25\).
   (A) The vertical component of the velocity of the ball.
   (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
   (C) The speed of the ball.
5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.

4 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .

1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~ms} { } ^ { 1 }\). The speed of projection is \(28 \mathrm {~ms} { } ^ { 1 }\).
2. Find the angle of projection of the stone.
3. Find the horizontal range of the stone.

2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(27 ^ { \circ }\).

4 A golfer hits a ball from a point \(O\) on horizontal ground with a velocity of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The horizontal range of the ball is \(R\) metres and the time of flight is \(t\) seconds.

1. Express \(t\) in terms of \(\theta\), and hence show that \(R = 125 \sin 2 \theta\). The golfer hits the ball so that it lands 110 m from \(O\).
2. Calculate the two possible values of \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_672_403_267_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A toy is constructed by attaching a small ball of mass 0.01 kg to one end of a uniform rod of length 10 cm whose other end is attached to the centre of the plane face of a uniform solid hemisphere with radius 3 cm . The rod has mass 0.02 kg , the hemisphere has mass 0.5 kg and the rod is perpendicular to the plane face of the hemisphere (see Fig. 1).

7. \begin{figure}[h] \end{figure} Figure 3 shows the path of a golf ball which is hit from the point \(O\) with speed \(49 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the ball is in a vertical plane containing \(O\) and the hole at which the ball is aimed. The hole is 170 m from \(O\) and on the same horizontal level as \(O\).

1. Suggest a suitable model for the motion of the golf ball. Find, correct to 3 significant figures,
2. the distance beyond the hole at which the ball hits the ground,
3. the magnitude and direction of the velocity of the ball when it is directly above the hole.

5. A firework company is testing its new brand of firework, the Sputnik Special. One of the company's employees lights a Sputnik Special on a large area of horizontal ground and it takes off at a small angle to the vertical. After a flight lasting 8 seconds it lands at a distance of 24 metres from the point where it was launched. The employee models the firework as a particle and ignores air resistance and any loss of mass which the Sputnik Special experiences. Using this model, find for this flight of the Sputnik Special,

1. the horizontal and vertical components of the initial velocity,
2. the initial speed, correct to 3 significant figures,
3. the maximum height attained.
4. Comment on the suitability of the modelling assumptions made by the employee.

5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).

1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
3. Find \(V\), in terms of \(u\).
4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).

5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-4_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).

1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
3. Find \(V\), in terms of \(u\).
4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).

7 An arrow is fired from a point \(A\) with a velocity of \(25 \mathrm {~ms} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. The arrow hits a target at the point \(B\) which is at the same level as the point \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-4_195_1093_1594_511}

1. State two assumptions that you should make in order to model the motion of the arrow.
   (2 marks)
2. Show that the time that it takes for the arrow to travel from \(A\) to \(B\) is 3.28 seconds, correct to three significant figures.
3. Find the distance between the points \(A\) and \(B\).
4. State the magnitude and direction of the velocity of the arrow when it hits the target.
5. Find the minimum speed of the arrow during its flight.

13

1. Show that $$h = 2.5 \sin ^ { 2 } \theta$$ 13 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) 13
2. Hence, given that \(0 ^ { \circ } \leq \theta \leq 60 ^ { \circ }\), find the maximum value of \(h\).
   13
3. Nisha claims that the larger the size of the ball, the greater the maximum vertical height will be. State whether Nisha is correct, giving a reason for your answer.

20 In this question use \(g = 9.8 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} Find \(a\)

\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.

1. Find an expression for the maximum height reached by the particle. [3]

\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.

1. Use the graph to find the value of \(u\). [2]
2. Find the time taken for the particle to return to ground level. [3]

A particle is projected vertically upwards with speed \(30\) m s\(^{-1}\) from a point on horizontal ground.

1. Show that the maximum height above the ground reached by the particle is \(45\) m. [2]
2. Find the time that it takes for the particle to reach a height of \(33.75\) m above the ground for the first time. Find also the speed of the particle at this time. [4]

A small ball is projected with speed \(15 \text{ m s}^{-1}\) at an angle of \(60°\) above the horizontal. Find the distance from the point of projection of the ball at the instant when it is travelling horizontally. [5]

A particle \(P\) is projected with speed \(26\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5\) m s\(^{-1}\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane. [4]
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(OA\). [3]

Two stones are projected simultaneously from a point \(O\) on horizontal ground. Stone \(A\) is thrown vertically upwards with speed \(98\) ms\(^{-1}\). Stone \(B\) is projected along the smooth ground in a straight line at \(24.5\) ms\(^{-1}\).

1. Find the distances of the two stones from \(O\) after \(t\) seconds, where \(0 \leq t \leq 20\). \hfill [3 marks]
2. Show that the distance \(d\) m between the two stones after \(t\) seconds is given by $$d^2 = 24.01(t^2 - 40t^2 + 425t^2).$$ \hfill [6 marks]
3. Hence find the range of values of \(t\) for which the distance between the stones is decreasing. \hfill [6 marks]

An object is projected vertically upwards with speed \(7\) m s\(^{-1}\). Calculate

1. the speed of the object when it is \(2.1\) m above the point of projection, [3]
2. the greatest height above the point of projection reached by the object, [3]
3. the time after projection when the object is travelling downwards with speed \(5.7\) m s\(^{-1}\). [3]

A particle is projected vertically upwards, from the ground, with a speed of \(28 \text{ m s}^{-1}\). Ignoring air resistance, find

1. the maximum height reached by the particle, [2]
2. the speed of the particle when it is 30 m above the ground, [3]
3. the time taken for the particle to fall from its highest point to a height of 30 m, [3]
4. the length of time for which the particle is more than 30 m above the ground. [2]

A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1. \includegraphics{figure_6_1}

1. When the man is descending with an acceleration 1.5 m s\(^{-2}\) downwards, how much time does it take for his speed to increase from 0.5 m s\(^{-1}\) downwards to 3.5 m s\(^{-1}\) downwards? How far does he descend in this time? [4]

The man has a mass of 80 kg. All resistances to motion may be neglected.

1. Calculate the tension in the wire when the man is being lowered
   1. with an acceleration of 1.5 m s\(^{-2}\) downwards,
   2. with an acceleration of 1.5 m s\(^{-2}\) upwards. [5]

Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.

1. For safety reasons, the tension in the wire should not exceed 2500 N. What is the maximum acceleration allowed when the man is being raised? [4]

At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N. \includegraphics{figure_6_2}

1. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the tension in the wire. [4]

A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O, as shown in Fig. 7. Air resistance is negligible. The initial horizontal component of the velocity of the firework is 21 m s\(^{-1}\).

1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is 28 m s\(^{-1}\). [4]
2. Show that the firework is \((28t - 4.9t^2)\) m above the ground \(t\) seconds after its projection. [1]

When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of 21 m s\(^{-1}\) and the other with a quarter of this speed.

1. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion. [3]
2. Find the distance between these parts of the firework
   1. when they reach the ground, [2]
   2. when they are 10 m above the ground. [5]
3. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac{4}{90}(120x - x^2)\), referred to the coordinate axes shown in Fig. 7. [4]

Anila is practising catching tennis balls. She uses a mobile computer-controlled machine which fires tennis balls vertically upwards from a height of 2.5 metres above the ground. Once it has fired a ball, the machine is programmed to move position rapidly to allow Anila time to get into a suitable position to catch the ball. The machine fires a ball at 24 ms\(^{-1}\) vertically upwards and Anila catches the ball just before it touches the ground.

1. Draw a speed-time graph for the motion of the ball from the time it is fired by the machine to the instant before Anila catches it. [3 marks]
2. Find, to the nearest centimetre, the maximum height which the ball reaches above the ground. [4 marks]
3. Calculate the speed at which the ball is travelling when Anila catches it. [4 marks]
4. Calculate the length of time that the ball is in the air. [3 marks]