(a) Show that the change in momentum over a time interval [to, t1] is equal to the integral of F from to to t1; that is, show that p(ti) – p(to) = " F(t) dt This integral is called the 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 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g, where g = 32 ft/s. (i) Find the change in the ball's momentum. (ii) Find the average force on the bat.

(a) Show that the change in momentum over a time interval [to, t1] is equal to the integral of F from to to t1; that is, show that p(ti) – p(to) = " F(t) dt This integral is called the 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 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g, where g = 32 ft/s. (i) Find the change in the ball's momentum. (ii) Find the average force on the bat.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

It may surprise you to learn that the collision of baseball and bat lasts only about a thou-
sandth 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 momentum p of an object is the product of its mass m and its velocity v, that is,
p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F(t)
that is a continuous function of time.
(a) Show that the change in momentum over a time interval [to, t1] is equal to the integral
of F from to to t1; that is, show that
p(t) – p(to) = |" F(t) dt
This integral is called the 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 s and leaves the bat with
velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is
measured in slugs: m = w/g, where g = 32 ft/s².
(i) Find the change in the ball's momentum.
(ii) Find the average force on the bat.

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