When given the values for the height and two distances for after and before the projectile touches the ground, but the angle is unknow and time unknown, how is the angle of projection found ?

To find the angle of projection in this scenario, you can use the following steps:

1. Use the given values for the height, initial and final distances to calculate the time of flight of the projectile. This can be done using the kinematic equation for vertical motion:

h = (1/2) * g * t^2

Where:
h = height of the projectile
g = acceleration due to gravity (approximately 9.81 m/s^2)
t = time of flight

2. Once you have calculated the time of flight, you can use it to find the horizontal velocity of the projectile. This can be done using the formula:

horizontal distance = horizontal velocity * time of flight

3. Now that you have the horizontal velocity, you can use it to find the angle of projection. The angle of projection can be calculated using the formula:

angle = arctan(vertical velocity / horizontal velocity)

Where:
vertical velocity = initial vertical velocity of the projectile

By following these steps, you should be able to find the angle of projection for the projectile given the height and two distances.