Projectiles: launch angle, launch height and clearing a wall Problem set A

A pebble is thrown horizontally at 8 m s⁻¹ from a height of 20 m above level ground.

1. (a) Find the time it takes to reach the ground.
2. (b) Find the horizontal distance from the launch point to where it lands.

A cricket ball is projected from the origin with speed 16 m s⁻¹ at 50° above the horizontal. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up.

1. (a) Write down the horizontal and the vertical component of the initial velocity.
2. (b) Write down the velocity vector at time \(t\) seconds.
3. (c) Write down the position vector at time \(t\) seconds.

A rugby ball is projected from level ground with speed 24 m s⁻¹ at 25° above the horizontal.

1. (a) Find the time it takes to reach its highest point.
2. (b) Find its greatest height.
3. (c) Find the horizontal distance from the launch point to where it lands.

An arrow is projected from a point 1.6 m above level ground with speed 30 m s⁻¹ at 20° above the horizontal.

1. (a) Find the time it takes to reach the ground.
2. (b) Find the horizontal distance from the launch point to where it lands.
3. (c) Find its greatest height above the ground.

A stone is thrown from a bridge. It is thrown horizontally at 6 m s⁻¹ and hits the water 2.5 seconds later. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up.

1. (a) Find its velocity as a column vector when it hits the water.
2. (b) Find its speed when it hits the water.
3. (c) Find the angle below the horizontal at which it hits the water.

A ball is kicked from level ground with speed 15 m s⁻¹ at 45° above the horizontal, towards a wall 4.5 m high that is 20 m away along the ground. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up, with the origin at the launch point.

1. (a) Find the time at which the ball is above the wall.
2. (b) Find the height of the ball at that time.
3. (c) Does the ball clear the wall? Give the difference between its height and the top of the wall.

A diver leaves horizontally at 3 m s⁻¹ from a point 12 m above level water.

1. (a) Find the time it takes to reach the water.
2. (b) Find how far from the point directly below the launch it lands.
3. (c) Find its speed and the angle below the horizontal as it reaches the water.

A javelin is released from a point 1.8 m above level ground with speed 28 m s⁻¹ at 35° above the horizontal. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up, with the origin on the ground below the release point.

1. (a) Write down the position vector at time \(t\) seconds.
2. (b) Find its greatest height above the ground.
3. (c) Find the time it takes to land and the horizontal distance of the landing point from the origin.

In a free kick the defenders form a wall. The ball is kicked from level ground with speed 20 m s⁻¹ at 25° above the horizontal. The wall of defenders is 1.9 m high and stands 9.15 m from the kick. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up, with the origin at the kick.

1. (a) Find the height of the ball when it is 9.15 m from the kick, and say whether it clears the 1.9 m wall.
2. (b) Find the speed of the ball at that moment.
3. (c) Find how far from the kick the ball lands.

A tennis ball is projected from level ground with speed 20 m s⁻¹ at 50° above the horizontal.

1. (a) Find the two times at which the ball is 6 m above the ground.
2. (b) Find the horizontal distance from the launch point at each of these times.
3. (c) For how long is the ball more than 6 m above the ground?

A ball is kicked from level ground at an angle of 45° above the horizontal, with speed \(V\) m s⁻¹, towards a wall 4 m high that is 20 m away along the ground. Take \(\mathbf{i}\) as horizontal and \(\mathbf{j}\) as vertically up, with the origin at the kick.

1. (a) Show that the height of the ball when it is 20 m from the kick is \(20 − \dfrac{3920}{V^{2}}\) metres.
2. (b) Find the least value of \(V\) for which the ball clears the wall.