Mechanics problem — One body thrown vertically and one thrown horizontally

[PeroK]

Maybe the textbook's solution is wrong

If that is the textbook solution in post #9, then you and I were right about the interpretation of the problem.

 

[PeroK]

Maybe the textbook's solution is wrong

The solution you posted in post #9 looks wromg to me. The $v_1^2$ term in the square root should be $-v_1^2$.

Note that for any $v_1$ the distance initially decreases. But, if $v_1$ is large, then the minimum distance occurs for small $t$. If $v_2$ is also large, then B soon passes under A and, again, the local minimum has passed.

This ties in with what I posted earlier about there being a maximum value of $v_1$ for a solution.

The other problem with the solution is that if the minimum occurs after B has hit the ground, then when B hits the ground is still a minumum for the allowed motion. That is to say: the distance decreases until B hits the ground (which is then the minimum), but it wouldn't be the minimum if B was allowed to fall further.

This leads to an inequality for $v_2$.

$$\frac{l}{2}\sqrt{\frac{g}{2h}} - (\frac{l^2g}{8h} - v_1^2)^{1/2} \le v_2 \le \frac{l}{2}\sqrt{\frac{g}{2h}} + (\frac{l^2g}{8h} - v_1^2)^{1/2} $$
Note that this also works in the degenerate case where $v_1 = 0$.

 

[haruspex]

If that is the textbook solution in post #9, then you and I were right about the interpretation of the problem.

And the question should have said the distance is "at its minimum just as..."

 

[neilparker62]

Is this the textbook's solution? Weird, my solution doesn't depend on $v_1$ at all, rather on $h,\,g$ and $x_B(t=0s)$.

I am obtaining a similar solution. There is constant relative velocity between the two projectiles.

 

[haruspex]

I am obtaining a similar solution. There is constant relative velocity between the two projectiles.

What interpretation of the question are you using? Based on the book solution I believe @PeroK's must be right, and I confirm his analysis in post #32.

 

[haruspex]

A at that time you calculated?

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