Projectile passing through given point

7 A particle is projected from the origin with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 2 seconds the particle is at a point which is 18 m horizontally from the origin and 4 m above it.

1. Show that \(\tan \theta = \frac { 4 } { 3 }\) and find \(u\).
2. Find the horizontal range of the particle.

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 with speed 35 m s\(^{-1}\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively. The equation of the trajectory of \(P\) is $$y = kx - \frac{(1 + k^2)x^2}{245},$$ where \(k\) is a constant. \(P\) passes through the points \(A(14, a)\) and \(B(42, 2a)\), where \(a\) is a constant.

1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is 63.435°, correct to 3 decimal places. [5]

For the larger angle of projection, calculate

1. the time after projection when \(P\) passes through \(A\), [2]
2. the speed and direction of motion of \(P\) when it passes through \(B\). [4]

A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]

It is given that \(\cos\theta = \frac{5}{9}\).

1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]

One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The particle \(P\) is held vertically below \(O\) with the string taut and then projected horizontally. When the string makes an angle of \(60°\) with the upward vertical, \(P\) becomes detached from the string. In its subsequent motion, \(P\) passes through the point \(A\) which is a distance \(a\) vertically above \(O\).

1. The speed of \(P\) when it becomes detached from the string is \(V\). Use the equation of the trajectory of a projectile to find \(V\) in terms of \(a\) and \(g\). [4]
2. Find, in terms of \(m\) and \(g\), the tension in the string immediately after \(P\) is initially projected horizontally. [4]

A particle \(P\) is projected with speed \(u\text{ms}^{-1}\) at an angle \(\tan^{-1}2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56\text{m}\) horizontally from \(O\), it is at a vertical height \(H\text{m}\) above the plane. When \(P\) has travelled a distance \(84\text{m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H\text{m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

A particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

\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]

Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35\text{ms}^{-1}\) at an angle of \(\alpha\) to the horizontal, where \(\cos\alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \((9.8t - 4.9t^2)\) m higher than O. When descending, the stone hits a plum which is \(3.675\) m higher than O. Air resistance should be neglected. Calculate the horizontal distance of the plum from O. [6]

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 gunner fires one shell from each of two guns on a stationary ship towards a vertical cliff \(AB\) of height \(100\) m whose foot \(A\) is at a horizontal distance \(600\) m from the point of projection.

1. Given that the shell from the first gun hits the cliff, travelling horizontally, at a point \(45\) m above \(A\), determine the initial velocity of the shell. Express your answer in the form \(a\mathbf{i} + b\mathbf{j}\), where \(a\) and \(b\) are integers. [6]
2. The shell from the second gun hits the cliff at its top point \(B\). Given that the initial speed of the shell is \(300\) m s\(^{-1}\), determine the possible angles of projection. [5]