Lecture 2.3: Newtons second law of motion Matthew Macauley - - PowerPoint PPT Presentation

Lecture 2.3: Newton’s second law of motion

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 1 / 6

Introduction

Newton’s 2nd law of motion

F = ma. (Force = mass × acceleration)

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 2 / 6

Falling objects with air resistance

Differential equation

We just derived the following ODE: v ′ = −g − r

mv, where v(t) is velocity,

g ≈ 9.81m/s2, m is mass, and r > 0 is a constant.

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 3 / 6

An example

Example 1 (exponential decay to a value)

A 70kg object falls from rest, and its terminal velocity is −20 m/s. (a) Find its velocity and distance traveled after 2 seconds. (b) How long does it take to reach 80% of its terminal velocity?

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 4 / 6

An example: v(t) = −20 + 20e− 1

20 gt

Example 1 (exponential decay to a value)

A 70kg object falls from rest, and its terminal velocity is −20 m/s. (a) Find its velocity and distance traveled after 2 seconds. (b) How long does it take to reach 80% of its terminal velocity?

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 5 / 6

An example: v(t) = −20 + 20e− 1

20 gt

Example 1 (exponential decay to a value)

A 70kg object falls from rest, and its terminal velocity is −20 m/s. (a) Find its velocity and distance traveled after 2 seconds. (b) How long does it take to reach 80% of its terminal velocity?

• M. Macauley (Clemson)

Lecture 2.3: Newton’s 2nd law of motion Math 2080, ODEs 6 / 6