3.02h Motion under gravity: vector form

\includegraphics{figure_11} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(U\) m s\(^{-1}\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.

1. Show that on impact the line of centres makes an angle of 30° with the vertical. [1]

The coefficient of restitution between A and B is \(\frac{1}{2}\). After the impact B moves freely under gravity.

1. Determine, in terms of \(U\) and \(g\), the time taken for B to first return to the table. [7]

In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}

1. Determine the following, in either order.
   [9]

After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.

1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]

A particle P of mass 1 kg is fixed to one end of a light inextensible string of length 0.5 m. The other end of the string is attached to a fixed point O, which is 1.75 m above a horizontal plane. P is held with the string horizontal and taut. P is then projected vertically downwards with a speed of \(3.2 \text{ m s}^{-1}\).

1. Find the tangential acceleration of P when OP makes an angle of \(20°\) with the horizontal. [2]

The string breaks when the tension in it is 32 N. At this point the angle between OP and the horizontal is \(\theta\).

1. Show that \(\theta = 23.1°\), correct to 1 decimal place. [5]

Particle P subsequently hits the plane at a point A.

1. Determine the speed of P when it arrives at A. [4]
2. Show that A is almost vertically below O. [5]

A small object, of mass 0.02 kg, is dropped from rest from the top of a building which is160 m high.

1. Calculate the speed of the object as it hits the ground. [3]
2. Determine the time taken for the object to reach the ground. [3]
3. State one assumption you have made in your solution. [1]

Points \(A\) and \(B\) lie on horizontal ground. At time \(t = 0\) seconds, an object \(P\) is projected from \(A\) towards \(B\) such that \(AB\) is the range of \(P\). The speed of projection is \(24 \cdot 5\) ms\(^{-1}\) in a direction which is 30° above the horizontal.

1. Calculate the range \(AB\) of the object \(P\). [5]

At time \(t = 1\) second, another object \(Q\) is projected from \(B\) towards \(A\) with the same speed of projection \(24 \cdot 5\) ms\(^{-1}\) and in a direction which is also 30° above the horizontal.

1. Determine the height above the ground at which \(P\) and \(Q\) collide. [5]

A tennis ball is projected with velocity vector \((30\mathbf{i} - 14\mathbf{j})\) ms\(^{-1}\) from a point \(P\) which is at a height of \(2.4\) m vertically above a horizontal tennis court. The ball then passes over a net of height \(0.9\) m, before hitting the ground after \(\frac{4}{7}\) s. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The origin \(O\) lies on the ground directly below the point \(P\). The base of the net is \(x\) m from \(O\). \includegraphics{figure_10}

1. Find the speed of the ball when it first hits the ground, giving your answer correct to one decimal place. [3]
2. After \(\frac{2}{5}\) s, the ball is directly above the net.
   1. Find the position vector of the ball after \(\frac{2}{5}\) s.
   2. Hence determine the value of \(x\) and show that the ball clears the net by approximately \(16\) cm. [4]
3. In fact, the ball clears the net by only \(4\) cm.
   1. Explain why the observed value is different from the value calculated in (b)(ii).
   2. Suggest a possible improvement to this model. [2]

\includegraphics{figure_13} A golfer hits a ball from a point \(A\) with a speed of \(25\text{ms}^{-1}\) 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 \(10\text{ms}^{-2}\). The ball first lands at a point \(B\) which is \(4\text{m}\) 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]

A girl is practising netball. She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop. The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.

• The initial velocity of the ball has magnitude \(U\) m s\(^{-1}\).
• The angle of projection is \(40°\).
• The ball is modelled as a particle.
• The hoop is modelled as a point.

This is shown on the diagram below. \includegraphics{figure_12}

1. For \(U = 10\), find
   1. the greatest height above the ground reached by the ball [5]
   2. the distance between the ball and the hoop when the ball is vertically above the hoop. [4]
2. Calculate the value of \(U\) which allows her to hit the hoop. [3]
3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl? [1]
4. Suggest one improvement that might be made to this model. [1]

A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).

1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]

\includegraphics{figure_11} A projectile is fired from a point \(O\) in a horizontal plane, with initial speed \(V\), at an angle \(\theta\) to the horizontal (see diagram).

1. Show that the range of the projectile on the horizontal plane is $$\frac{2V^2 \sin \theta \cos \theta}{g}.$$ [4]

There are two vertical walls, each of height \(h\), at distances 30 m and 70 m, respectively, from \(O\) with bases on the horizontal plane. The value of \(\theta\) is \(45°\).

1. If the projectile just clears both walls, state the range of the projectile. [1]
2. Hence find the value of \(V\) and of \(h\). [5]

A stone is projected vertically upwards from ground level at a speed of \(30\,\mathrm{m}\,\mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{~m} \mathrm{~s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{m} \mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]

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]