Question

While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is 3.54 kg · m2 and its rotational kinetic energy is 167 J.
(a)
What is the angular velocity of the leg? (Give your answer in rad/s.)
(b)
What is the velocity (in m/s) 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)

Answer 1

(a) Use K = 1/2 I ω^2 so ω = sqrt(2K/I) = sqrt(2·167 / 3.54) = sqrt(94.35) ≈ 9.71 rad/s.

(b) v_tip = ω r = 9.71 · 1.05 m = 10.2 m/s (approximately).

(c) The ball can leave faster than the instantaneous speed of the shoe because of how momentum and energy are transferred during the contact:
- For an (approximately) elastic collision between a heavy striker (shoe+leg) and a much lighter ball, the lighter object can rebound with a speed larger than the striker’s initial speed (for a perfectly elastic hit and M >> m, the ball can approach 2·v_shoe).
- In a real kick the shoe and ball deform and the shoe/leg can continue to do positive work on the ball during the finite contact time (muscles keep accelerating the leg/shoe through the contact), so the ball can gain more energy and exit faster than the shoe’s instantaneous speed at first contact. Both effects allow the ball to achieve a speed greater than the shoe tip’s initial velocity.