Question

a wheeled car with a spring-loaded cannon fires a metal ball vertically. if the vertical initial speed of the ball is 5 m/s as the cannon moves horizontally at a speed of .75 m/s. how far from the launch point does the ball fall back into the cannon? what would happen if the cannon were accelerating

Answer 1

reference to the ground, at the beginning the cannon ball contains a vertical velocity (5m/s) as well as horizontal velocity of (.75 m/s). You can verify this by applying relative motion theories on to this problem ( to the horizontal direction). Now this will become a simple trajectory problem.

apply,$$\uparrow S=ut+\frac 12 at^2$$ and find the time taken to travel, use the same equation horizontally to find the vertical distant traveled by the ball.

Answer 2

along the vertical axis, v = u + at (assume gravity of 10 m/ss)
0 = 5 - 10t (ie cannon ball stops at time t)
t = 0.5s (ie til cannon ball stops rising)
time for round trip = 2t = 1s
distance travelled by cannon = vt = 0.75*1 = 75cm.

if the cannon is accelerating, then it will race ahead of the cannonball.

you would need to apply v = u + at and other derivative equations in the horizontal axis to calculate how far it would race ahead. the cannonball itself would stop accelerating as soon as it breaks contact with the cannon.