How Do Projectile Angles Affect Velocity at a Fixed Height?

[merzperson]

Homework Statement

The three balls in the figure which have equal masses, are fired with equal speeds at the angles shown. Rank in order, from largest to smallest, their speeds as they cross the dashed horizontal line. (All three are fired with sufficient speed to reach the line.)

Homework Equations

E_p = mgh
K = 0.5mv²

The Attempt at a Solution

Since the v_o is equal for each ball, we can conclude that v_ocy > v_oby > v_oay. Since the force of gravity has the same effect on each ball, we can conclude that at the line v_c > v_b > v_a

However, I am getting the problem wrong on MasteringPhysics. I know this is an energy problem, but I can't get past how simple the problem is in my head.

 

[ehild]

How much will be the kinetic energy of each ball at the horizontal line if they start with equal speeds and rise at equal heights?

ehild

 

[ideasrule]

Forget physics. Forget the horizontal component of the velocity too; that's irrelevant to the balls' vertical motion. If you throw a ball in the air, do you think it's going to be faster at, say, a height of 2 m if you throw it up really fast, or if you throw it up really slowly?

 

[merzperson]

@ideasrule:
I'm not sure what you're getting at. If you throw a ball up really fast, the velocity at h=2m will be greater than if you throw it up really slow. How does this apply to the above problem and how does it help to prove that my attempt at a solution is not valid? If anything it seems like it further established my original solution.

@ehild:
I'm not sure I understand what you are getting at either. Each ball starts with equal speed and makes it to the height of the line, but they do not make it to the same final height. This means that the kinetic energy at the line will be different for each ball.

Thank you both for your replies! However, I still do not see how the solution can't be vc > vb > va

EDIT:
I have solved the problem. I found out that I was ignoring the horizontal velocity at the line, so my original answer was correct but was the answer to a different question. The total velocity (not only in the y direction) of the three balls at the line were equal because of conservation of energy (I believe). Since K=0.5mv² and the masses are the same, and the initial velocities are the same, the velocities at the line must also be the same.

Thanks guys for your help.