It may surprise you to learn that the collision of a baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.

The of an object is the product of its mass and its velocity , that is, . Suppose an object, moving along a straight line, is acted on by a force that is a continuous function of time.

(a) Show that the change in momentum over a time interval is equal to the integral of from to ; that is, show that

This integral is called the of the force over the time interval.

(b) A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 and leaves the bat with velocity 110 mi/h. A baseball weights 5 oz and, in US Customary units, its mass is measured in slugs: where .

(i)Find the change in the ball’s momentum.

(ii)Find the average force on the bat.