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?
To determine under which condition the ball will fall back into the chimney of a stationary toy train while it is moving, we need to consider the motion of the train and the properties of the ball.
When the train is moving, it is undergoing both translational and rotational motion. The ball shot by the spring gun will inherit the initial velocity of the train when it is first expelled from the chimney.
To ensure that the ball falls back into the chimney, we need to consider the vertical motion of the ball. Gravity will act on the ball, causing it to accelerate downwards. We need to make sure that the ball's initial velocity upwards is less than the magnitude of the downward velocity due to gravity.
The condition for the ball to fall back into the chimney is that the initial velocity of the ball in the vertical direction must be less than or equal to the velocity with which the ball is shot into the air, minus the velocity due to gravity.
Mathematically, we can express this condition as:
V_initial_vertical ≤ V_shot - g
Where:
- V_initial_vertical represents the initial vertical velocity of the ball (upwards).
- V_shot represents the magnitude of the velocity at which the ball is shot into the air.
- g represents the acceleration due to gravity.
By calculating V_initial_vertical, we can determine whether the ball will fall back into the chimney or not.