The chimney of a stationary toy train consists of a vertical spring gun that shoots steel balls a meter or so straight into the air – so straight that the ball always falls back into the chimney.
If the train is moving, under which condition will the ball fall back into the chimney?
moving with constant velocity (slow enough that air resistance is not a factor)
When the train is moving, the ball will fall back into the chimney if its initial vertical velocity is exactly equal to the vertical velocity of the train. This condition ensures that the ball will have the same vertical displacement as the moving train by the time it reaches its peak height and starts to fall back down.
To understand this condition more clearly, let's break it down step by step:
1. Consider the situation from an observer standing on the ground. From their perspective, the ball is shot straight up into the air and will follow a purely vertical path due to the force from the spring gun.
2. Now, since the train is moving, it will also have a vertical velocity of its own. Let's assume that the train is moving in the positive direction, which means its velocity is upward.
3. When the ball leaves the chimney, it will initially have the same upward velocity as the train, since they are both part of the same system.
4. As the ball rises, it will experience gravity pulling it downward, causing it to decelerate. At the same time, the train's velocity remains constant.
5. The ball will reach its peak height when its upward velocity becomes zero. At this point, it starts to fall back down.
6. For the ball to fall back into the chimney, its downward velocity due to gravity must match the upward velocity of the train exactly. This ensures that the ball falls back into the chimney and does not overshoot.
So, in summary, the ball will fall back into the chimney when its initial upward velocity matches the upward velocity of the train. This condition ensures that the ball reaches the same height as the moving train and falls back down in sync with it.