Modeling C10M.1 A 42-g racquetball flying at 21 m/s exactly horizon- tally slams

Question

Modeling C10M.1 A 42-g racquetball flying at 21 m/s exactly horizon- tally slams into a wall. Midway between the instant that the racquetball begins to touch the wall and the instant when the ball leaves the wall after it rebounds, there is an instant where the ball is at rest, flattened against the wall. Model the racquetball as if it were a spring. (a) Carefully explain why we can treat the ball as an iso- lated system throughout the process of rebounding from the wall even though it clearly participates in a number of significant external interactions. (b) Calculate the racquetball's effective spring constant if at the instant it is at rest, its horizontal width (= the equivalent spring's "length") is half its normal value of 5.7 cm. (Assume that all of the increase in the ball's internal energy goes to spring energy.)

Explanation / Answer

a) We can treat the ball as an isolated system on the basis on conservation of energy. As the racquetball hits the wall at some velocity, its kinetic energy gets converted to spring energy. As we assume that there is no loss in energy during the impact.

b) As per isolated system, we have

0.5*M*V2 = 0.5*k*x2

where k is the spring constant and x is the compression, M is mass of ball and V is the velocity with which it strikes the wall.

Therefore, k = MV2 / x

Just put all the given values in S.I units, We get

k = 0.042 kg * 212 / 0.0285

k = 649 N/m

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