(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A ball is thrown straight upward with an initial speed v0. When it reaches

the top of its flight at height h, a second ball is thrown straight upward with

the same initial speed. Do the balls cross paths (a) at height 1⁄2h (b) above

1⁄2h or (c) below 1⁄2h. Explain your reasoning.

3. The attempt at a solution

My approach was quite simplistic. Since the second ball has an initial speed, over the given time interval, it'll move over a longer distance than the first one. That was also what the answers said. However, I don't find this answer satisfying. I would like to prove this analytically but unfortunately I can't find a way to do so.

I tried to write out the distance equations for both balls:

[tex]

h=v_0t - \frac{gt^2}{2}

[/tex]

[tex]

h=h_t - \frac{gt^2}{2}

[/tex]

where [tex]h[/tex] is the height at which the balls cross paths, [tex]h_t[/tex] is the max height the first ball reaches and [tex]v_0[/tex] is the initial speed. Now the RHS's of both equations are equal, so equating them and solving for t gives

[tex]

t=\frac{h_t}{v_0}

[/tex]

Substituting that back into the first of the two equations gives

[tex]

h=h_t-\frac{gh_t^2}{2v_0^2}

[/tex]

From here, I can't get any further. I'm not even sure this is the right approach.

Thank you for any help.

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# Homework Help: Simple kinematics ball problem

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