∙Tennis, anyone? Tennis players sometimes leap into the air...

$\bullet$ Tennis, anyone? Tennis players sometimes leap into the air to return a volley. (a) If a 57 g tennis ball is traveling horizontally at 72 $\mathrm{m} / \mathrm{s}$ (which does occur), and a 61 kg tennis player leaps vertically upward and hits the ball, causing it to travel at 45 $\mathrm{m} / \mathrm{s}$ in the reverse direction, how fast will her center of mass be moving horizontally just after hitting the ball? (b) If, as is reasonable, her racket is in contact with the ball for $30.0 \mathrm{ms},$ what force does her racket exert on the ball? What force does the ball exert on the racket?

Key Concepts

Conservation of Linear Momentum

This principle states that in the absence of an external net force, the total momentum of a system remains constant. In collision problems, it is used to relate the momentum of objects before and after an interaction, allowing one to solve for unknown velocities or masses in situations where no external forces act along the direction of interest.

Impulse-Momentum Theorem

This theorem links the impulse applied to an object (the product of force and the time interval over which it is applied) to the change in momentum of that object. It is particularly useful in collision scenarios, as it provides a way to calculate the average force exerted during the very short time interval of the collision.

Newton's Third Law

This law states that for every action, there is an equal and opposite reaction. In the context of collisions, it indicates that the force exerted by one object on another is met with an equal and opposite force. This is critical when analyzing interactions between two bodies, such as a tennis racket and a tennis ball, ensuring that the forces experienced by each are equal in magnitude and opposite in direction.

Transcript

00:01 Now we have a tennis player of mass 61 kilograms and a tennis ball of mass 57 grams that comes towards the player at 72 meter per second.

00:14 The player jumps and hits the ball mid -air and the ball goes back with velocity 45 meter per second.

00:23 Now we want to find out the velocity with which the person is now moving, the tennis player is now moving backwards.

00:31 This happens because the person hits the ball.

00:36 And applies force on it in the leftward direction to change its velocity and the reactionary force is on the player in the rightward direction which causes him to accelerate in the rightward direction and he starts moving.

00:53 We know that there are no other external forces in the x direction and hence the total momentum should remain a constant, which means the change in momentum must be zero, which means the change in the momentum of the ball must be summed with the change in the momentum of the tennis player.

01:15 Must be 0.

01:16 Hence, if we simply find the change in the momentum of the ball, we find the change in the momentum of the player.

01:25 We know that the initial momentum of the ball is 57 grams, that is 57 times 10 power minus 3 times the velocity 72 meter per second, but if we choose our x -axis to be in the right -word direction, this would be a minus 72.

01:41 And the final momentum would be 57 times 10 power minus 3 times 45 meter per second with a positive sign, because i'm sorry, this would be plus 72 and this would be minus 45.

01:55 Because 45 is in the left -word direction, in the negative x -axis direction.

02:01 Now, delta p of the ball is simply the final minus initial momentum, which is 57 times 10 power minus 3 into minus 45, minus 57 into 10 power minus 3 into 72.

02:20 Now that we know this, this is 51 times 10 power minus 3.

02:25 Minus 3 multiplied by minus 117.

02:30 Now that we know this, we use this equation...

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