Tennis balls traveling at greater than 100 mph routinely bounce off tennis racke

Question

Tennis balls traveling at greater than 100 mph routinely bounce off tennis rackets. At some sufficiently high speed, however, the ball win break through the strings and keep going. Suppose that a 100 g tennis ban traveling at 200 mph is just sufficient to break the 2.0 mm thick strings. Model the racket as a potential energy barrier of width L = 2.0 mm whose height is the energy of the slowest string-breaking ball. If a tennis ball approaches the racket from the left at 120 mph. estimate the probability that it will tunnel through the racket without breaking the strings. Give your answer as a power of 10 rather than a power of e.

Explanation / Answer

one mile = 1609.344 m
maximum speed v = 200 mph                             = (200)(1609.344 / 3600) m/s                             = 89.4 m/s let a ball (with speed v = 200 mph) is just sufficient to break the strings ,          energy U = (1/2)mv2                        = (1/2)(100*10-3 kg)(89.4 m/s)2                        = 399.618 J tennis ball speed v' = 120 mph                               = (120)(1609.344 / 3600) m/s                               = 53.64 m/s kinetic energy E = (1/2)(100*10-3 kg)( 53.64 m/s)2                          = 143.88 J the penetration distance is             = (1 / 2){h / 2m(U - E)}                 = (1 / 2){6.63*10-34 J.s / [(2)(0.1 kg)(399.618 J - 143.88 J)]}                 = 1.48*10-35 m tunneling probability is           P = e-2L/ apply log on both sides ,we get       logP = (-2L/ )loge        logP = (-(2)(2*10-3 m)/1.48*10-35 m){loge}        logP = [- 2.7*1032 ]{loge} therefore ,         logP = - 1.17*1032        P = (10)- 1.17*10^32              

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