A small steel ball B is at rest on the edge of a table of height h=1m. Another steel ball A, used as the bob of a simple pendulum also of length h=1m, is released from rest with the pendulum suspension horizontal, and swings against B as shown in the diagram.The masses of the balls are identical and the collision is elastic.
(a)Which ball is in motion for the longest time?
(b)Which ball covers the greatest distance?
To determine which ball is in motion for the longest time and which ball covers the greatest distance, we need to analyze the motion of the balls after the collision.
First, let's consider the collision between the two balls. Since the collision is elastic (meaning there is no loss of kinetic energy), the kinetic energy is conserved. Therefore, the initial kinetic energy of ball A before the collision is transferred to ball B after the collision and vice versa.
(a) Which ball is in motion for the longest time?
After the collision, ball A will start moving in the horizontal direction with its initial kinetic energy. This horizontal motion will continue until it reaches the edge of the table, where it will fall vertically downwards due to gravity. The time it takes for ball A to reach the edge of the table is equal to the time it takes for the ball B to return to its initial position (since they have equal kinetic energy).
However, since the table is at a height h=1m, the time it takes for ball B to fall from the edge of the table to the ground will be longer than the time it takes for ball A to reach the edge of the table horizontally. Therefore, ball A will be in motion for the longest time.
(b) Which ball covers the greatest distance?
Since the length of the pendulum is also h=1m, ball A will swing in a circular motion. The distance covered by ball A in one complete swing (or one period) is equal to the circumference of the circle formed by the pendulum's swing. The circumference of a circle is given by C = 2πr, where r is the radius of the circle.
In this case, the radius of the circle is also h=1m. Therefore, the distance covered by ball A in one complete swing is 2π(1) = 2π meters.
On the other hand, ball B is only moving back and forth horizontally until it reaches the edge of the table, covering a distance equal to the horizontal distance from its initial position to the edge of the table.
Comparing the two distances, 2π meters covered by ball A in one complete swing is greater than the horizontal distance covered by ball B. Therefore, ball A covers the greatest distance.
In summary:
(a) Ball A is in motion for the longest time.
(b) Ball A covers the greatest distance.