Finding out angle of projection given only vertical and horizontal motions?

To find the angle of projection given only the vertical and horizontal motions, you can use the following steps:

1. Determine the initial velocity components in the horizontal and vertical directions. This can be done by using the equations of motion:

- Horizontal motion: \(v_{0x} = v_0 \cos(\theta)\)
- Vertical motion: \(v_{0y} = v_0 \sin(\theta)\)

Where:
\(v_{0x}\) = initial velocity in the horizontal direction
\(v_{0y}\) = initial velocity in the vertical direction
\(v_0\) = initial velocity of the projectile
\(\theta\) = angle of projection

2. Use the vertical motion equation to find the time of flight (\(t\)):

\(y = v_{0y}t - \frac{1}{2}gt^2\)

Where:
\(y\) = vertical displacement
\(g\) = acceleration due to gravity

3. Use the horizontal motion equation to find the range (\(R\)):

\(R = v_{0x}t\)

4. Use the range and time of flight to find the angle of projection:

\(\theta = \arctan(\frac{y}{R})\)

By following these steps, you can determine the angle of projection given only the vertical and horizontal motions of the projectile.