3.02i Projectile motion: constant acceleration model

A golf ball is hit with initial velocity \(u\) ms\(^{-1}\) at an angle of \(45°\) above the horizontal. The ball passes over a building which is \(15\) m tall at a distance of \(30\) m horizontally from the point where the ball was hit.

1. Find the smallest possible value of \(u\). [7 marks]

When \(u\) has this minimum value,

1. show that the ball does not rise higher than the top of the building. [4 marks]
2. Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
3. Briefly describe two modelling assumptions that you have made. [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]

A particle is projected horizontally with a speed of 6 m s\(^{-1}\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground. [6]

A particle is projected with speed 49 m s\(^{-1}\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac{x^2(1 + \tan^2 \theta)}{490}.$$ [4]

\includegraphics{figure_8} The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta_1\) and \(\theta_2\), and the corresponding points where the particle returns to the plane are \(A_1\) and \(A_2\) respectively (see diagram).

1. Find \(\theta_1\) and \(\theta_2\). [4]
2. Calculate the distance between \(A_1\) and \(A_2\). [5]

A particle is projected with speed \(u\) ms\(^{-1}\) at an angle of \(\theta\) above the horizontal from a point \(O\). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and \(\theta\) and hence obtain the equation of trajectory $$y = x \tan \theta - \frac{gx^2 \sec^2 \theta}{2u^2}.$$ [4]

In a shot put competition, a shot is thrown from a height of 2.1 m above horizontal ground. It has initial velocity of 14 ms\(^{-1}\) at an angle of \(\theta\) above the horizontal. The shot travels a horizontal distance of 22 m before hitting the ground.

1. Show that \(12.1 \tan^2 \theta - 22 \tan \theta + 10 = 0\), and find the value of \(\theta\). [5]
2. Find the time of flight of the shot. [2]

A particle is projected horizontally with a speed of \(7 \text{ ms}^{-1}\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground. [6]

A small ball of mass 0.2 kg is projected with speed \(11 \text{ ms}^{-1}\) up a line of greatest slope of a roof from a point \(A\) at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at \(B\). The roof is rough and the coefficient of friction is \(\frac{1}{4}\). The distance \(AB\) is 5 m and \(AB\) is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \text{ ms}^{-1}\), correct to 2 decimal places. [6]

The ball leaves the roof at \(B\) and moves freely under gravity. The point \(C\) is at the lower edge of the roof. The distance \(BC\) is 5 m and \(BC\) is inclined at \(30°\) to the horizontal.

1. Determine whether or not the ball hits the roof between \(B\) and \(C\). [7]

A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).

1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]

Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.

1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]

The coefficient of restitution between the two particles is \(\frac{1}{2}\).

1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]

Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}

1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
   1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
   2. Calculate the work done against friction per tile. [2]
   3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]

A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}

1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
2. Find the distance \(PQ\). [7 marks]

A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]

A ship \(A\) has maximum speed 30 km h\(^{-1}\). At time \(t = 0\), \(A\) is 70 km due west of \(B\) which is moving at a constant speed of 36 km h\(^{-1}\) on a bearing of 300°. Ship \(A\) moves on a straight course at a constant speed and intercepts \(B\). The course of \(A\) makes an angle \(\theta\) with due north.

1. Show that \(-\arctan \frac{4}{3} \leq \theta \leq \arctan \frac{4}{3}\). [7]
2. Find the least time for \(A\) to intercept \(B\). [5]

\includegraphics{figure_2} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45°\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Find the magnitude of the impulse which acts on \(A\) in the collision. [9]

\includegraphics{figure_3} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.

1. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall. [5]

At 12 noon, ship \(A\) is 20 km from ship \(B\), on a bearing of \(300°\). Ship \(A\) is moving at a constant speed of 15 km h\(^{-1}\) on a bearing of \(070°\). Ship \(B\) moves in a straight line with constant speed \(V\) km h\(^{-1}\) and intercepts \(A\).

1. Find, giving your answer to 3 significant figures, the minimum possible for \(V\). [3]

It is now given that \(V = 13\).

1. Explain why there are two possible times at which ship \(B\) can intercept ship \(A\). [2]
2. Find, giving your answer to the nearest minute, the earlier time at which ship \(B\) can intercept ship \(A\). [8]

A ship \(A\) is travelling at a constant speed of 30 km h\(^{-1}\) on a bearing of \(050°\). Another ship \(B\) is travelling at a constant speed of \(v\) km h\(^{-1}\) and sets a course to intercept \(A\). At 1400 hours \(B\) is 20 km from \(A\) and the bearing of \(A\) from \(B\) is \(290°\).

1. Find the least possible value of \(v\). (3)

Given that \(v = 32\),

1. find the time at which \(B\) intercepts \(A\). (8)

A cyclist starting from rest accelerates uniformly at \(1.5 \text{ m s}^{-2}\) for \(4\) s and then travels at constant speed.

1. Sketch a velocity-time graph to represent the first \(10\) seconds of the cyclist's motion. [2]
2. Calculate the distance travelled by the cyclist in the first \(10\) seconds. [2]

\includegraphics{figure_11} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertical axis \(Oy\). \(P\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).

1. Determine, in terms of \(U\), \(V\) and \(g\), the distance \(OC\). [4] \includegraphics{figure_11b} \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\). [2]
3. Hence determine an expression for \(d\) in terms of \(U\), \(V\), \(g\) and \(h\). [3]
4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{1}{2}\), determine an expression for \(V\) in terms of \(g\), \(d\) and \(h\). [4]

\includegraphics{figure_11} A golfer hits a ball from a point \(A\) with a speed of 25 m s\(^{-1}\) at an angle of 15° above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be 10 m s\(^{-2}\). The ball first lands at a point \(B\) which is 4 m below the level of \(A\) (see diagram).

1. Determine the time taken for the ball to travel from \(A\) to \(B\). [3]
2. Determine the horizontal distance of \(B\) from \(A\). [2]
3. Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball. [4]

The horizontal distance from \(A\) to \(B\) is found to be greater than the answer to part (b).

1. State one factor that could account for this difference. [1]

A small ball \(B\) moves in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertically upwards axis \(Oy\). \(B\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U \mathrm{m s}^{-1}\) and \(V \mathrm{m s}^{-1}\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.

1. Show that the path of \(B\) has equation \(2U^2 y = 2UVx - gx^2\). [3]

During its motion, \(B\) just clears a vertical wall of height \(\frac{1}{3}a\) m at a horizontal distance \(a\) m from \(O\). \(B\) strikes the ground at a horizontal distance \(3a\) m beyond the wall.

1. Determine the angle of projection of \(B\). Give your answer in degrees correct to 3 significant figures. [5]
2. Given that the speed of projection of \(B\) is \(54.6 \mathrm{m s}^{-1}\), determine the value of \(a\). [2]
3. Hence find the maximum height of \(B\) above the ground during its motion. [3]
4. State one refinement of the model, other than including air resistance, that would make it more realistic. [1]

In this question you should take the acceleration due to gravity to be \(10 \text{ms^{-2}\).} \includegraphics{figure_12} A small ball \(P\) is projected from a point \(A\) with speed \(39 \text{ms}^{-1}\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac{5}{13}\) and \(\cos \theta = \frac{12}{13}\). Point \(A\) is \(20 \text{m}\) vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.

1. Find the maximum height of \(P\) above the ground during its motion. [3]

The time taken for \(P\) to travel from \(A\) to \(C\) is \(7\) seconds.

1. Determine the value of \(T\). [3]
2. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]

At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geq 0\), the velocity \(v \text{ms}^{-1}\) of \(Q\) is given by $$v = kt^3 + 6t^2 + \frac{3}{2}t,$$ where \(k\) is a positive constant.

1. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision. [6]

In this question use \(g = 9.81\) m s\(^{-2}\) A particle is projected with an initial speed \(u\), at an angle of 35° above the horizontal. It lands at a point 10 metres vertically below its starting position. The particle takes 1.5 seconds to reach the highest point of its trajectory.

1. Find \(u\). [3 marks]
2. Find the total time that the particle is in flight. [3 marks]

A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal. The ball needs to first land at a point at least \(d\) metres away from \(P\). You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2\theta \geq \frac{dg}{u^2}$$ [6 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) A toy shoots balls upwards with an initial velocity of 7 m s\(^{-1}\) The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground.

1. Suppose that the toy shoots the balls vertically upwards.
   1. Verify the claim in the advertisement. [2 marks]
   2. State two modelling assumptions you have made in verifying this claim. [2 marks]
2. In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical. The range of maximum heights, \(h\) metres, above the ground which can be reached by the balls may be expressed as $$k \leq h \leq 2.5$$ Find the value of \(k\) [4 marks]