While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is $3.75\text{ kg}\cdot {\text{m}}^2$ and its rotational kinetic energy is 175 J. (a) What is the angular velocity of the leg? (b) What is the velocity of tip of the punter’s shoe if it is 1.05 m from the hip joint? (c) Explain how the football can be given a velocity greater than the tip of the shoe (necessary for a decent kick distance).

Video Transcript

This is College Physics Answers with Shaun Dychko. A kicker is going to punch a football and the kicker's leg has a moment of inertia of 3.75 kilogram meters squared and a rotational kinetic energy of 175 joules and our job is to find out what is the angular speed of their foot given this information. So rotational kinetic energy is one-half times moment of inertia times angular speed squared and then we can multiply both sides by 2 and divide both sides by I to solve for ω squared and then take the square root of both sides and we are left with ω; it's the square root of 2 times kinetic energy divided by moment of inertia. So that's 2 times 175 joules divided by 3.75 kilogram meters squared all square rooted which is 9.66 radians per second. The speed of a point on the end of this rotating leg is going to be the radius times the angular speed. So that's 1.05 meters times its angular speed which is 10.1 meters per second. And part (c) says well how come given that the foot is moving only 10.1 meters per second, why is it that the football will be moving faster than that? And it's not speed that's conserved in a collision, it's momentum that is conserved in a collision and the momentum of the ball, after collision, is approximately going to be the same as the momentum of the foot before the collision. So we can substitute for each of these: mass of the ball times the velocity of the ball after collision approximately is the same as mass of the foot times velocity of the foot— before collision— and solve for v ball prime and we see that it's going to be the initial velocity of foot multiplied by the ratio of the mass of the foot to the mass of the ball. Now since the foot has a much greater mass than the ball, this fraction is going to be a number greater than 1 and so that's going to make this velocity of the ball after collision greater than the initial velocity of the foot before hitting the football.