Trajectory

Question

In a very popular lecture demonstration, a project is fired at the falling target as an Figure P3 .59. The projectile leaves the gun at the same instant that the target is dropped from rest. Assuming that the gun is initially aimed at the target, show that the projectile will hit the target. (when restriction of this experiment is that the project down my speech the target before the target strikes the floor.)

Answer 1

Coordinates of the bullet with respect to the origin: (v₀cos(A)t, v₀sin(A)t - ½gt²)

 

Coordinates of falling target located at a horizontal distance, D, and height, H, with respect to the origin: (D, H - ½gt²)

 

 

The gun will be aimed at the target, so the initial angle of the bullet is related to D & H:

 

tan(A) = H/D

 

 

The bullet will hit the falling target if its height is the same as the height of the target when the bullet reaches the horizontal distance, D.

 

When the bullet reaches D, D = v₀cos(A)t  and  t = D/(v₀cos(A)).

 

 

y_bullet = v₀sin(A)D/(v₀cos(A)) - ½gD²/v₀²cos²(A)

 

= Dtan(A) - gD²/2v₀²cos²(A) = H -gD²/2v₀²cos²(A)

 

 

y_target = H - gD²/2v₀²cos²(A)

 

 

So, y_bullet = y_target if the bullet is fast enough to travel a distance D before it reaches the ground.