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 m v , where v ( t ) is velocity, g ≈ 9 . 81 m / s 2 , 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 + 20 e − 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 + 20 e − 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