Will your answer change if the cart is accelerating in the forward direction? if so, how?

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Yes. The answer changes if the cart is accelerating.

As shown above, after the ball is fired the only external force on the ball is the weight acing vertically. There are no forces acting in the horizontal direction. So the net acceleration is in the vertical direction and the vertical velocity changes but the horizontal velocity of the ball stays constant at .

Let's say the ball takes a time to come back down. If the cart is accelerating in the forward direction, the velocity of the cart at time is greater than .

At time , the horizontal velocity of the ball is still . Hence relative to the cart, the horizontal velocity of the ball is smaller than the velocity of the cart .

So when the ball comes back down, it will land behind the launching tube.