(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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc056aa58-bf9c-4143-8824-c19a449b7bda%2F72904432-4c9c-46d1-8b6b-c2872c10abc7%2Fgv4ofh_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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