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.)
Calculus and Baseball Applied Project 1 from section 6.5 of Calculus: Early Transcendentals by James Stewart...
1. In Major League Baseball, a typical fastball is pitched at a speed of about 90 mph, or 40 m/s. The batter can perform the remarkable task of completely reversing the direction of this ball and sending it back at about 105 mph, or 47 m/s. Doing this at just the right millisecond to get the ball to go where the batter wants it to go is such an amazing feat that a Yale physicist once jokingly declared that hitting...
(12) 12. A baseball with mase 145 12. A baseball with mass 145 g is thrown to the left (negative direction) by a Houston Astros pitcher at 96 mph (42.9 m/s) and is smacked by a Washington Nationals batter for a home run. (Treat as a 1-D problem by ignoring the y component) a) Find the initial momentum of the baseball, including sign. b) If the impulse delivered by the bat is +10.8 kg m/s, find the final velocity of...
Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force ΣF⃗ acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse J⃗ is a measure of both these effects. For a constant net force, the impulse is given by J⃗ =F⃗ Δt. The impulse is a vector pointing in the same direction as the force vector. The units...
Multivariable Calculus help with the magnitude of angular momentum: My questions is exercise 4 but I have attached exercise 1 and other notes that I was provided 4 Exercise 4. In any mechanics problem where the mass m is constant, the position vector F sweeps out equal areas in equal times the magnitude of the angular momentum ILI is conserved (Note: be sure to prove "if and only if") (Note: don't try to use Exercise 2 in the proof of...