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**ReneePatt** 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 $\displaystyle momentum$ $\displaystyle p$ of an object is the product of its mass $\displaystyle m$ and its velocity $\displaystyle v$, that is, $\displaystyle p=mv$. Suppose an object, moving along a straight line, is acted on by a force $\displaystyle F=F(t)$ that is a continuous function of time.

(a) Show that the change in momentum over a time interval $\displaystyle [t_{0},t_{1}]$ is equal to the integral of $\displaystyle F$ from $\displaystyle t_{0}$ to $\displaystyle t_{1}$; that is, show that

$\displaystyle p(t_{1})-p(t_{0})=\int_{t_0}^{t_1}F(t) dt$

This integral is called the $\displaystyle impulse$ 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 $\displaystyle s$ 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: $\displaystyle m=\frac{w}{g}$ where $\displaystyle g=32 ft/s^2$.

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

(ii)Find the average force on the bat.