A tennis ball of mass mt is held just above a basketball of...

A tennis ball of mass $m_{t}$ is held just above a basketball of mass $m_{b},$ as shown in Figure P8.22. With their centers vertically aligned, both are released from rest at the same moment so that the bottom of the basketball falls freely through a height $h$ and strikes the floor. Assume an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down because the balls have separated a bit while falling. Next, the two balls meet in an elastic collision. (a) To what height does the tennis ball rebound? (b) How do you account for the height in (a) being larger than $h$ ? Does that seem like a violation of conservation of energy?

Key Concepts

Elastic Collisions

This concept refers to collisions where both kinetic energy and momentum are conserved. In an elastic collision, not only is the total momentum of the objects maintained, but the kinetic energy before and after the collision remains unchanged as well. This is crucial for analyzing how velocities change during impacts—particularly when multiple collisions occur in sequence—while ensuring no energy is "lost" to deformation or heat.

Conservation of Momentum

The conservation of momentum states that in an isolated system, the total momentum remains constant if no external forces act on it. This principle is essential when analyzing collisions between objects, as it allows us to relate the velocities and masses of the bodies before and after impacts. It plays a significant role in understanding the transfer of energy and momentum in sequential collisions.

Conservation of Mechanical Energy

In systems where only conservative forces (like gravity) do work, the total mechanical energy is conserved. When objects undergo free fall or participate in elastic collisions, the potential energy converts into kinetic energy and vice versa. This concept is key to explaining how an object can rebound to a height which seems to exceed the original drop height, as energy is redistributed within the system rather than created from nothing.

Free Fall Kinematics

Free fall kinematics deals with the motion of objects under the influence of gravity alone, assuming negligible air resistance. It explains how an object converts gravitational potential energy into kinetic energy as it falls. Understanding free fall is essential for determining the speeds of objects just prior to collisions, which in turn affects the outcomes of subsequent elastic collisions.

Transcript

00:08 We have mass of tennis ball m t equals to 57 gram and mass of basketball mb equals to 590 gram okay and h equals to 1 .2 meter so now for the part a for the basketball earth system we can apply conservation of energy as delta k plus delta u equals to zero or k final minusia initial plus u final minus u initial equals to zero okay so now since basketball is released from rest and finally reaches the ground we can write 1 by 2 m b b square final minus 0 plus 0 minus m b g h equals to okay.

01:09 So from here we get vb final equals to under root 2 g h.

01:15 So now substituting the values we get under root 2 multiplied by g is 9 .8 and h is 1 .2 meter.

01:23 So from here we get vb final equals to 4 .85 meter per second in downward direction.

01:31 Okay.

01:33 So now now the initial speed, initial velocity of the basketball after the collision with the ground will be vb initial equals to 4 .85 meter per second and it will be upward and the initial velocity of the tennis ball vti will be equals to minus 4 .85 meter per second in the downward direction that's why it is negative okay so for the for the elastic collision we can determine the velocity of tennis ball final equals to 2 m b divided by m t plus m b multiplied by v b initial plus m t minus m b divided by m t plus m t plus m t plus m b multiplied by v b v t initial okay v t initial so now substituting all the values we get v tn is final equals to two multiplied by mb is 590 grams so 0 .59 kilogram divided by m10s is 57 gram so 0 .057 program plus 0 .59 multiplied by vb initial is 4 .85 plus substituting the values 0 .057 minus 0 .59 divided by 0 .057 plus 0 .59 multiplied by minus 4 .85.

03:20 Okay.

03:21 So after solving this value 4 .85.

03:24 So after solving this value, v tennis final comes out to be 12 .84 meter per second.

03:33 So now applying conservation of energy for the tennis ball earth system, we can write k initial plus u initial equals to k final plus u final...

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