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

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SUMMARY

The relationship between velocity and displacement in a falling object is illustrated through the equations tan θ = vy/vx and tan θ = sy/sx. The first equation is used to determine the angle of impact, which is derived from the velocity vector just before the object strikes the ground. In contrast, the second equation relates to the overall displacement from the cliff to the impact point. Understanding the distinction between these two equations is crucial for accurately analyzing projectile motion, particularly in parabolic trajectories.

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

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

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