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 h1, h2 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 vy1(0) = -u·sinθ and vx1(0) = u·cosθ, so from Eqs.(2) we have B1 = -u·sinθ and D1=u·cosθ. Then from the conditions x1(0) = 0 and y1(0) = h1, we have C1 = 0 and A1 = h1. Thus Eqs.(2) become

The initial conditions for bullet 2 are y2(0) = h2, x2(0) = L, vy2(0) = u·sinθ, and vx2(0) = -u·cosθ. Thus Eqs.(3) become

For two bullets to collide at time τ, we need to have y1(τ) = y2(τ) and x1(τ) = x2(τ). From Eqs.(2A) and (3A) we have

They become, after applying Eqs.(1),

The above two equations are identical if h1 ≠ h2 and they give the same τ as they should. If h1 = h2, then sinθ = 0 and the first equation holds for any τ, implying that the two bullets are at the same height at any moment since both of them initially do not have any vertical speed and both of them fall from the same height; it is the second equation that gives a definite τ in this case. Any way we have

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