Professor Lewin releases the ball from his waist with zero motion to ensure a predictable trajectory. The ball's velocity at its lowest point can be determined using the principle of conservation of energy. The ball will reach a height of 1.2 m on the other side, assuming no energy loss. Professor Lewin's careful release minimizes the risk of accidents, and due to energy loss, the maximum heights for each swing will gradually decrease.
a. To determine the ball's velocity at its lowest point, we can use the principle of conservation of energy. Before the ball is released, it has potential energy given by E_before = mgh, where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the initial height (1.2 m).
After the ball is released, at its lowest point, all of its potential energy is converted into kinetic energy. Therefore, we have E_after = 1/2mv^2, where v is the velocity of the ball at its lowest point.
Setting the two equations equal to each other (E_before = E_after), we have mgh = 1/2mv^2. We can cancel out the mass, and solve for v: v = sqrt(2gh).
b. To determine how high the ball would go on the other side, we can use the principle of conservation of mechanical energy. The ball's mechanical energy remains constant throughout its motion, neglecting any energy loss to heat or air resistance.
At the highest point of the swing, all of the ball's initial potential energy is converted into gravitational potential energy. Therefore, the ball's height on the other side can be determined using the equation mgh = mgh', where h' is the final height.
Since the ball is released from a height of 1.2 m, the final height h' would be equal to 1.2 m.
c. Professor Lewin can be confident that he won't be going to the hospital because he releases the ball from his waist, ensuring that the ball has zero motion at position A. This means that the ball's velocity is zero when it is released. By releasing the ball with zero velocity, Professor Lewin ensures that it follows a predictable trajectory and avoids any unexpected collisions or dangerous movements.
d. Professor Lewin is careful about how he releases the ball to ensure that it has zero motion when released. By doing so, he guarantees that the ball follows a consistent and controlled path, minimizing the risk of accidents or injuries. Releasing the ball with a non-zero velocity could lead to unpredictable movements and potential hazards.
e. Since the ball will lose a tiny bit of energy due to factors such as air resistance and friction, the maximum heights for each swing will decrease over time. The ball's energy is gradually dissipated as it undergoes multiple swings, resulting in a gradual decrease in the maximum height it reaches on subsequent swings. This loss of energy over time is why the ball's maximum heights will be slightly lower with each swing.