**Problem 1C-08**:** Two-dimensional two gun problem without air resistance

In Problem 1A-07 two identical guns are lined up vertically and are fired
simultaneously. In this problem two guns are lined up with an angle to the
vertical line as shown in the picture. Under what condition will two bullets
collide? If two bullets do collide, find the time and the position of the
collision. Consider h_{1}, h_{2} and L in the figure as given,
and let the initial speed of the bullets be u. A bullet after the firing is a
free thrown object. Use the position functions as given in Problem 1C-07 as
the functions for free thrown objects.

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**Solution:**

From the picture we can see that

The position and the velocity of bullet No.1 are, from Problem 1C=07,

The position and the velocity of bullet No.2 are

The initial speed of bullet 1 is v

The initial conditions for bullet 2 are y

For two bullets to collide at time τ, we need to have y

They become, after applying Eqs.(1),

The above two equations are identical if h

The position of collision, (X, Y), is obtained from

However, Y must be zero or positive, otherwise two bullets will collide under the ground. The condition Y ≥ 0 leads to

In summary, for two bullets to collide at time τ at position (X, Y), they must be

under the constraint

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