3.02i Projectile motion: constant acceleration model

13 A cricket ball is hit from a point \(P\) on a sloping field. The initial velocity of the ball is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above the field, which under the path of the ball slopes upwards at \(10 ^ { \circ }\) to the horizontal. Air resistance is to be ignored.

1. Find the vertical height of the ball above the field after 2.5 seconds.
2. The ball lands on the field at the point \(X\). Find the distance \(P X\).

12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

9 A particle is projected from a point \(O\) on horizontal ground with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal.

1. Write down the equation of the trajectory, in terms of \(\tan \theta\).
2. The particle passes through a point whose horizontal and vertical distances from \(O\) are 72 m and \(y \mathrm {~m}\) respectively. By considering the equation of the trajectory as a quadratic equation in \(\tan \theta\), or otherwise, find the greatest possible value of \(y\).

7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.

1. Find the initial speed of the particle.
2. Find the magnitude and direction of the velocity of the particle when it hits the ground.

7 A particle is projected with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle 0.4 s after projection.

12 A particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

12 A particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
3. A plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.

14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).

1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$

4 A particle is projected with velocity \(V\), at an angle of elevation of \(60 ^ { \circ }\) to the horizontal, from a point on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The path of the particle is in a vertical plane through a line of greatest slope. If \(R _ { 1 }\) and \(R _ { 2 }\) are the respective ranges when the particle is projected up the plane and down the plane, show that $$R _ { 2 } = 2 R _ { 1 }$$

A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \text{ m s}^{-1}\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).

1. Using the equation of the particle's trajectory and the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), show that the possible values of \(\tan \theta\) are \(\frac{4}{3}\) and \(\frac{1}{4}\). [4]
2. Find the distance \(OA\) for each of the two possible values of \(\tan \theta\). [3]
3. Sketch in the same diagram the two possible trajectories. [2]

A stone is projected from a point \(O\) on horizontal ground. The equation of the trajectory of the stone is $$y = 1.2x - 0.15x^2,$$ where \(x\) m and \(y\) m are respectively the horizontal and vertically upwards displacements of the stone from \(O\). Find

1. the greatest height of the stone, [2]
2. the distance from \(O\) of the point where the stone strikes the ground. [2]

\includegraphics{figure_4} A small ball \(B\) is projected from a point \(O\) above horizontal ground, with initial speed \(15\) m s\(^{-1}\) at an angle of projection of \(30°\) above the horizontal (see diagram). The ball strikes the ground \(3\) s after projection.

1. Calculate the speed and direction of motion of the ball immediately before it strikes the ground. [5]
2. Find the height of \(O\) above the ground. [2]

A small ball is projected with speed \(16 \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal from a point on horizontal ground. Calculate the period of time, before the ball lands, for which the speed of the ball is less than \(12 \text{ ms}^{-1}\). [4]

A particle is projected at an angle of \(θ°\) below the horizontal from a point at the top of a vertical cliff \(26 \text{ m}\) high. The particle strikes horizontal ground at a distance \(8 \text{ m}\) from the foot of the cliff \(2 \text{ s}\) after the instant of projection. Find

1. the speed of projection of the particle and the value of \(θ\), [6]
2. the direction of motion of the particle immediately before it strikes the ground. [3]

A particle is projected at an angle of \(θ°\) below the horizontal from a point at the top of a vertical cliff \(26\) m high. The particle strikes horizontal ground at a distance \(8\) m from the foot of the cliff \(2\) s after the instant of projection. Find

1. the speed of projection of the particle and the value of \(θ\), [6]
2. the direction of motion of the particle immediately before it strikes the ground. [3]

A particle is projected with speed \(20\,\text{m}\,\text{s}^{-1}\) at an angle of \(60°\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40°\) below the horizontal. [4]

A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10\,\text{m}\,\text{s}^{-1}\) horizontally and \(15\,\text{m}\,\text{s}^{-1}\) vertically. 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 hence find the equation of the trajectory of the particle. [4]
2. The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff. Show that \(d\) is less than \(30\). [2]
3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]

A particle is projected with speed \(20 \text{ ms}^{-1}\) at an angle of \(60°\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40°\) below the horizontal. [4]

A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \text{ ms}^{-1}\) horizontally and \(15 \text{ ms}^{-1}\) vertically. 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 hence find the equation of the trajectory of the particle. [4]
2. Show that \(d\) is less than \(30\). [2]
3. Find the value of \(x\) when the particle is \(14\) m below the level of \(O\). [2]

The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d\) m from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.

A small ball \(B\) is projected from a point \(O\) on horizontal ground. The initial velocity of \(B\) has horizontal and vertically upwards components of \(18 \text{ ms}^{-1}\) and \(25 \text{ ms}^{-1}\) respectively. For the instant \(4 \text{ s}\) after projection, find the speed and direction of motion of \(B\). [4]

A small object is projected from a point \(O\) with speed \(V \text{ ms}^{-1}\) at an angle of \(45°\) above the horizontal. At time \(t\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the path. [4]

The object passes through the point with coordinates \((24, 18)\).

1. Find \(V\). [2]
2. The object passes through two points which are \(22.5 \text{ m}\) above the level of \(O\). Find the values of \(x\) for these points. [3]

\includegraphics{figure_1} A small ball \(B\) is projected from a point \(O\) on horizontal ground towards a point \(A\) 12 m above the ground. 0.9 s after projection \(B\) has travelled a horizontal distance of 20 m and is vertically below \(A\) (see diagram).

1. Find the angle and the speed of projection of \(B\). [4]
2. Calculate the distance \(AB\) when \(B\) is vertically below \(A\). [2]

A particle \(P\) is projected from a point \(O\) on horizontal ground with initial speed 20 m s\(^{-1}\) and angle of projection 30°. At time \(t\) s after projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of \(P\). [4]
2. Calculate this height. [3]

\(P\) is at the same height above the ground at two points which are a horizontal distance apart of 15 m.

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]