How to find the Eg of an object thrown at an angle?

[nilfound]

Homework Statement

Four identical balls are thrown from the top of a cliff, each with the same speed. The first is thrown straight up, the second is thrown at 30° above the horizontal, the third at 30° below he horizontal, and the fourth straight down. How do the speeds and kinetic energies of the balls compare as they strike the ground…

1. when air resistance is negligible?
2. when air resistance is significant?

Homework Equations

Eg = mgh, Ek = 1/2mv^2
Et = Ek where height <= 0
Et = Eg where height > 0

The Attempt at a Solution

Eg = mghcos()?

 

[Jilang]

All the balls start with the same speed from the same height, so have the same energy at that point.

 

[nilfound]

But when they hit the ground, the angle they were thrown at will have a significant difference in their speed as gravity grabs hold right? So won't the first ball, in the extreme case of gained height, have a greater final velocity and kinetic energy? Or am I missing a fundamental part of energy conservation :p

 

[nilfound]

Because there are two heights: the starting height at the top of the cliff, and the ending height at the bottom. I am just confused on how to calculate how they will be different or the same!

 

[nilfound]

For energy it will be the same of course! But the velocities will vary will they not? But then if the velocity varies, then Ek changes, because Ek = 1/2mv^2!

 

[Jilang]

Velocity can have a vertical and a horizontal component.

 

[berkeman]

How do the speeds and kinetic energies of the balls compare as they strike the ground…

Is that the exact wording of the question? Or is it maybe asking about the vertical component of the velocity when the balls hit the ground?

 

[berkeman]

Also, can you list the relevant kinematic equations of motion for a constant acceleration (gravitational acceleration downward)? Those will be helpful in solving this problem.