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

A marble is thrown horizontally at 1.5 m s⁻¹ from a height of 1.2 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 football is projected from the origin with speed 30 m s⁻¹ at 35° 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 basketball is projected from level ground with speed 11 m s⁻¹ at 60° 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.

A ball is projected from a point 5 m above level ground with speed 12 m s⁻¹ at 35° 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 pebble is thrown from a pier. It is thrown horizontally at 9 m s⁻¹ and hits the water 1.8 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 football is kicked from level ground with speed 18 m s⁻¹ at 40° above the horizontal, towards a net 5.2 m high that is 22 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 net.
2. (b) Find the height of the ball at that time.
3. (c) Does the ball clear the net? Give the difference between its height and the top of the net.

A skier leaves horizontally at 11 m s⁻¹ from a point 8 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 shot put is released from a point 2.1 m above level ground with speed 13 m s⁻¹ at 38° 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.

A school team practises kicking over a fence. The ball is kicked from level ground with speed 17 m s⁻¹ at 32° above the horizontal. The fence is 2.4 m high and stands 12 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 12 m from the kick, and say whether it clears the 2.4 m fence.
2. (b) Find the speed of the ball at that moment.
3. (c) Find how far from the kick the ball lands.

A golf ball is projected from level ground with speed 35 m s⁻¹ at 40° above the horizontal.

1. (a) Find the two times at which the ball is 10 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 10 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 6 m high that is 30 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 30 m from the kick is \(30 − \dfrac{8820}{V^{2}}\) metres.
2. (b) Find the least value of \(V\) for which the ball clears the wall.