What is the relationship between velocity and displacement in a falling object?

[smb360rewind]

Just worked through the "ball-falls-off-a-cliff" problem and was curious as why the equation
$$tan \theta = \frac{vy}{vx}$$ is used to find the angle at which the ball strikes the ground versus $$tan \theta = \frac{sy}{sx}$$

Homework Statement

Homework Equations

The Attempt at a Solution

 

[TSny]

Draw a sketch of the path of the ball from the cliff to the ground. The trajectory is a curve (parabolic). The "direction that the ball strikes the ground" is the direction of the velocity vector just before impact. Can you see that the velocity vector at impact does not have the same slope as the straight line drawn from the point where the ball leaves the cliff to the point of impact?

 

[smb360rewind]

Thank you. Overlaying the two equations in a sketch reveals your point. I'm starting to understand the importance of "sketching" problems.