3.02h Motion under gravity: vector form

\includegraphics{figure_3} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \((-20\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\). Immediately after being struck it has velocity \((15\mathbf{i} + 16\mathbf{j})\) m s\(^{-1}\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\). [4]
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\). [6]
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\). [4]
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic. [2]

\includegraphics{figure_3} The object of a game is to throw a ball \(B\) from a point \(A\) to hit a target \(T\) which is placed at the top of a vertical pole, as shown in Figure 3. The point \(A\) is 1 m above horizontal ground and the height of the pole is 2 m. The pole is at a horizontal distance of 10 m from \(A\). The ball \(B\) is projected from \(A\) with a speed of 11 m s\(^{-1}\) at an angle of elevation of \(30°\). The ball hits the pole at the point \(C\). The ball \(B\) and the target \(T\) are modelled as particles.

1. Calculate, to 2 decimal places, the time taken for \(B\) to move from \(A\) to \(C\). [3]
2. Show that \(C\) is approximately 0.63 m below \(T\). [4]

The ball is thrown again from \(A\). The speed of projection of \(B\) is increased to \(V\) m s\(^{-1}\), the angle of elevation remaining \(30°\). This time \(B\) hits \(T\).

1. Calculate the value of \(V\). [6]
2. Explain why, in practice, a range of values of \(V\) would result in \(B\) hitting the target. [1]

\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
3. find the distance \(BD\). [7]

\includegraphics{figure_3} [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector \(47.5\mathbf{j}\) metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \((2u\mathbf{i} + 5u\mathbf{j})\) m s\(^{-1}\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30\mathbf{i}\) metres, as shown in Figure 3.

1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s. [6]
2. Find the value of \(u\). [2]
3. Find the speed of \(P\) at \(B\). [5]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((6\mathbf{i} + 12\mathbf{j})\) m s\(^{-1}\), and passes through the point \(A\) at time \(t\) seconds after projection. The point \(B\) is on the horizontal plane vertically below \(A\), as shown in Figure 3. It is given that \(OB = 2AB\). Find

1. the value of \(t\), [7]
2. the speed, \(V\) m s\(^{-1}\), of the ball at the instant when it passes through \(A\). [5]

At another point \(C\) on the path the speed of the ball is also \(V\) m s\(^{-1}\).

1. Find the time taken for the ball to travel from \(O\) to \(C\). [3]

\includegraphics{figure_4} A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9a\) from \(O\) and at a height \(6a\) above the level of \(O\). The ball is projected with speed \(\sqrt{(27ag)}\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.

1. Show that tan\(^2 \theta - 6\) tan \(\theta + 5 = 0\) [7]

The two possible angles of projection are \(\theta_1\) and \(\theta_2\), where \(\theta_1 > \theta_2\).

1. Find tan \(\theta_1\) and tan \(\theta_2\). [3]

The particle is projected at the larger angle \(\theta_1\).

1. Show that the time of flight from \(O\) to \(T\) is \(\sqrt{\left(\frac{78a}{g}\right)}\). [3]
2. Find the speed of the particle immediately before it hits \(T\). [3]

\includegraphics{figure_1} The points \(O\) and \(B\) are on horizontal ground. The point \(A\) is \(h\) metres vertically above \(O\). A particle \(P\) is projected from \(A\) with speed 12 m s\(^{-1}\) at an angle \(\alpha°\) to the horizontal. The particle moves freely under gravity and hits the ground at \(B\), as shown in Figure 1. The speed of \(P\) immediately before it hits the ground is 15 m s\(^{-1}\).

1. By considering energy, find the value of \(h\). [4]

Given that 1.5 s after it is projected from \(A\), \(P\) is at a point 4 m above the level of \(A\), find

1. the value of \(\alpha\), [3]
2. the direction of motion of \(P\) immediately before it reaches \(B\). [3]

\includegraphics{figure_2} A particle is at the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(u\), where \(u < \sqrt{ag}\). The particle leaves the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.

1. Find an expression for \(\cos \theta\) in terms of \(u\), \(g\) and \(a\). [7]

The particle strikes the plane with speed \(\sqrt{\frac{9ag}{2}}\).

1. Find, to the nearest degree, the value of \(\theta\). [7]

A stone is dropped from rest at a height of 7 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards with half the speed with which it hit the ground. Calculate

1. the time taken for the stone to fall to the ground, [2 marks]
2. the speed with which the stone hits the ground, [2 marks]
3. the height to which the stone rises before it comes to instantaneous rest. [3 marks]

State two modelling assumptions that you have made. [2 marks]

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]

A particle of mass \(0.1\) kg is at rest at a point \(A\) on a rough plane inclined at \(15°\) to the horizontal. The particle is given an initial velocity of \(6\) m s\(^{-1}\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after \(1.5\) s.

1. Find the coefficient of friction between the particle and the plane. [7]
2. Show that, after coming to instantaneous rest, the particle moves down the plane. [2]
3. Find the speed with which the particle passes through \(A\) during its downward motion. [6]

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 stone is released from rest on a bridge and falls vertically into a lake. The stone has velocity \(14\text{ m s}^{-1}\) when it enters the lake.

1. Calculate the distance the stone falls before it enters the lake, and the time after its release when it enters the lake. [4]

The lake is \(15\text{ m}\) deep and the stone has velocity \(20\text{ m s}^{-1}\) immediately before it reaches the bed of the lake.

1. Given that there is no sudden change in the velocity of the stone when it enters the lake, find the acceleration of the stone while it is falling through the lake. [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 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]

Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,

1. calculate the time it would take for the stone to reach the sea, [3 marks]
2. find the speed with which the stone would hit the water. [2 marks]

Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is \(u\) m s\(^{-1}\) and it hits the stone before they both reach the water.

1. Find the minimum value of \(u\). [5 marks]
2. If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer. [2 marks]

\includegraphics{figure_2} Figure 2 shows a particle \(A\) of mass 5 kg, lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg. The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,

1. show that the magnitude of the force exerted on the pulley is \(\frac{10\sqrt{5}}{7}\) g N. [7 marks]
2. find the speed with which \(A\) hits the pulley. [3 marks]

When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.

1. Find the speed with which \(B\) hits the ground. [4 marks]

Andrew hits a tennis ball vertically upwards towards his sister Barbara who is leaning out of a window 7.5 m above the ground to try to catch it. When the ball leaves Andrew's racket, it is 1.9 m above the ground and travelling at \(21 \text{ m s}^{-1}\). Barbara fails to catch the ball on its way up but succeeds as the ball comes back down. Modelling the ball as a particle and assuming that air resistance can be neglected,

1. find the maximum height above the ground which the ball reaches. [4 marks]
2. find how long Barbara has to wait from the moment that the ball first passes her until she catches it. [6 marks]

\includegraphics{figure_4} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30°\) and \(60°\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.

1. Find the acceleration of \(P\) down the slope. [3 marks]
2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. [5 marks] \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \text{ m s}^{-2}\),
3. find, correct to 3 significant figures, the value of \(\mu\). [5 marks] In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
4. Find the value of \(t\) correct to 2 decimal places. [4 marks]

Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H\) m directly above A. \includegraphics{figure_5} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4\text{ms}^{-1}\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V\text{ms}^{-1}\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). [7]

In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of 8 ms\(^{-1}\) in a direction making an angle of 30° with the horizontal. \includegraphics{figure_8}

1. Find the time taken by the ball to travel 6 m horizontally. [2 marks]
2. Showing your method clearly, decide whether or not the ball will hit the coconut. [4 marks]
3. Find the greatest height reached by the ball above the level from which it was thrown. [4 marks]
4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of 8 m s\(^{-1}\). [3 marks]
5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut. [2 marks]

Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.

1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
2. Find the time for which the ball is in flight. [4 marks]
3. Find the horizontal distance travelled by the ball. [3 marks]
4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]

A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after 3.3 seconds.

1. Calculate the height above ground from which the lead was dropped. [2 marks]

According to a simple model, the ball hits the ground at the same time as the lead.

1. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. [2 marks]

The piece of lead is now thrown again from \(P\), with speed 7 ms\(^{-1}\) at an angle of 30° to the horizontal, as shown. \includegraphics{figure_6}

1. Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown. [4 marks]
2. Deduce that \(y = \frac{\sqrt{3}}{3}x - \frac{2}{15}x^2\). [3 marks]
3. Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\). [5 marks]

An aeroplane, travelling horizontally at a speed of 55 ms\(^{-1}\) at a height of 600 metres above horizontal ground, drops a sealed packet of leaflets. Find

1. the time taken by the packet to reach the ground, [3 marks]
2. the horizontal distance moved by the packet during this time. [2 marks]

The packet will split open if it hits the ground at a speed in excess of 125 ms\(^{-1}\).

1. Determine, with explanation, whether the packet will split open. [5 marks]
2. Find the lowest speed at which the aeroplane could be travelling, at the same height of 600 m, to ensure that the packet will split open when it hits the ground. [3 marks]

One of the leaflets is stuck to the front of the packet and becomes detached as it leaves the aeroplane.

1. If the leaflet is modelled as a particle, state how long it takes to reach the ground. [1 mark]
2. Comment on the model of the leaflet as a particle. [2 marks]