Lecture 17: Galileo, Newton and Baseball

This paragraph introduces the application of calculus in understanding the motion of objects, particularly in projectile motion. It discusses the historical context, mentioning Galileo's experiments with falling bodies and Newton's laws of motion and universal gravitation. The lecturer aims to explain how calculus can describe the path and velocity of projectiles, including cannonballs, baseballs, and planets. The paragraph also refutes Aristotle's claim that heavier bodies fall faster than lighter ones, using Galileo's experiments as evidence that all bodies fall at the same rate in the absence of air resistance.

The second paragraph delves into Newton's formulation of the law of universal gravitation, which posits that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. It also touches on Newton's anecdotal inspiration from a falling apple. The paragraph outlines Newton's three laws of motion, which include Galileo's principle of inertia, the relationship between force, mass, and acceleration (F=ma), and the principle of action and reaction. Additionally, it discusses Kepler's laws of planetary motion, particularly the elliptical orbits of planets, and how Newton's laws could explain these observations.

This paragraph presents a simplified mathematical model for the motion of a falling body under the influence of gravity, assuming constant acceleration. It explains that an object dropped near the Earth's surface will accelerate at approximately 32 feet per second squared, directed downward. The paragraph further illustrates how to calculate the velocity and position of a falling object over time, using basic calculus concepts. The讲师 also clarifies that this model is an approximation that holds true for objects falling over short distances near the Earth's surface, where gravitational acceleration can be considered constant.

The fourth paragraph extends the discussion to the motion of a thrown baseball, considering both its vertical and horizontal components. It provides an example where a ball is thrown with an initial vertical velocity of 48 feet per second and a horizontal velocity of 100 feet per second. The讲师 uses calculus to derive the equations for the ball's velocity and position over time, ultimately determining where and when the ball will land. This analysis is crucial for understanding how outfielders in baseball can predict the landing point of a fly ball.

In this paragraph, the lecturer explores the practical application of calculus in baseball, specifically for outfielders determining their position to catch a fly ball. It explains how the outfielder can use the angle of the ball's trajectory and the rate at which the angle changes to predict the landing point. The讲师 demonstrates that the key is for the outfielder to stand at a position where the slope of the line of sight to the ball changes at a constant rate, indicating the ball's landing spot. The paragraph also recounts the famous 'The Catch' by Willie Mays during the 1954 World Series as an example of such calculations in action.

The final paragraph presents a theoretical scenario where a baseball game is played in an environment where the Earth's mass is concentrated at its center, resulting in a gravitational field that could cause a projectile to follow an elliptical orbit. In this hypothetical situation, an outfielder would not need to calculate the ball's landing spot but could simply wait for the ball to orbit back to the point of impact. The lecturer humorously concludes that baseball is not played in such a theoretical Earth, where there is no land to play on, and looks forward to the next lecture on computing distances traveled by moving objects.