Physics

Problem

A tennis ball of mass

58\,\text{g}

is approaching a stationary racquet horizontally at

10\,\text{m\,s}^{-1}

(the positive

x

–direction is defined to be the direction the ball is moving just before impact). While the ball is in contact with the strings, the magnitude of the net horizontal force exerted on the ball varies with time as follows: • from

t=0

t=4\,\text{ms}

the force increases uniformly from

0\,\text{N}

600\,\text{N}

• from

t=4\,\text{ms}

t=6\,\text{ms}

the force remains constant at

600\,\text{N}

• from

t=6\,\text{ms}

t=8\,\text{ms}

the force decreases uniformly back to

0\,\text{N}

Assume no other horizontal forces act on the ball during this interval. (a) Describe qualitatively how the impulse delivered to the ball builds up during each of the three time intervals indicated above. (b) Identify the direction of the total impulse on the ball and relate it to the direction of the ball’s initial momentum. (c) Explain how a comparison of the magnitudes of the total impulse and the ball’s initial momentum indicates whether or not the ball reverses direction after leaving the racquet. (d) Without performing any explicit calculations, rank the magnitudes of the impulses delivered during the three individual time intervals and justify your ranking.

Intuitive explanation

This problem is about impulse and momentum change — how a time-varying force reverses the ball's motion. Rather than tracking the detailed acceleration at each instant, you can find the total momentum change by calculating the total impulse, which is the area under the force–time graph. Since the graph is made of simple geometric shapes (triangle, rectangle, triangle), you can find the total impulse without calculus, then use $\Delta p = J$ to find the ball's final velocity.

Intuitive explanation

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