Question

Calculus and Baseball Applied Project 1 from section 6.5 of Calculus: Early Transcendentals by James Stewart due: May 14, 2020, by 11:59 pm In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especialy the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Rober Adair, The Physics of Baseball, 3rd ed. (New York, 2002). 1. It may surprise you to learn that the collision of baseball and bat only lasts about a thousandth of a second. Here we calculate the average force on the bat 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. 1 (a) Show that the change in momentum over the time interval [t0, t1] is equal to the integral of F from t0 to t1; that is, show that p(t1) p(t0) = Z t1 t0 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. (Instructor’s hint: You must change the units so that all velocities are in ft/s and weights are in pounds.)

Answer 1

Given data : Tnitial velocity of baseball - go m/h 90 X 5280 60 X 60 ft sec V 3-132_ft/sec Final velocity of baseball 110 in/h 110 x 5280 3600 ft Sec V = 161.3 ft/sec 2 Inass of baseball - W 9 5 oz 32 5 -4 = 97.65 x 10 Jb 16X32 time of contact = oool sec il Change in ball's momentum = m (V-V 97.66 X10 + [1613 -3.+132) = 97.66.X.10-4 [293.37 2.86_pounds ft/sec AP = 2.86 lb ft/sec AP tit Edt: Impulse Ex. 0.001= AP EE 4p 0.001 2.86 0.001 F: 2860_lb_f1/52