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
