## Saturday, March 31, 2012

Blog Entry #67

Introduction

The goal is to find the velocity of a ball, and the angle it travels, given the initial velocity and angle.

The next two tables are for reference. Thanks to my cousin Gina for helping me with the Coefficient of Restitution! You are the best!

Variables:

Given:
v_i = initial velocity
θ = angle of impact before the bounce

Want to find:
v_f = final velocity
Φ = angle of travel after the ball bounces

Assumptions:
1. Gravity has negligible impact. We are talking about the moment just before the ball bounces to just after the ball bounces.
2. The coefficient of resistution (COR) is measured experimentally (see table). I thank my awesome cousin Gina Ramirez helped me with getting data. (see table above)
3. The x-direction is "forwards/backwards" and the y-direction is "up towards the sky/down towards the earth"
4. The coefficeint of friction only affects the y-direction (is this correct?)
5. Kinetic energy is not conserved.
6. The ground does not move and maintains a zero velocity.

Variables:
COR = coefficient of restitution, depending on the type of ball and surface the ball is being bounced on
μ = coefficient of friction, dependent on the surface the ball is being bounced on
v_ix, v_iy = initial velocity of the ball in the x-direction and y-direction, respectively
v_fx, v_fy = final velocity of the ball in the x-direction and y-direction, respectively

COR = √ ( (height of ball after the bounce) / (height of the ball before the bounce) )

Since momentum must be conserved in both the x and y directions:

m * v_i * cos θ = m * v_f * cos Φ
m * v_i * sin θ = m * v_f * sin Φ

m * v_i * cos θ - m * v_f * cos Φ = 0
m * v_i * sin θ - m * v_f * sin Φ = 0

m * v_i * cos θ - m * v_f * cos Φ = m * v_i * sin θ - m * v_f * sin Φ

m * ( v_i * cos θ - v_f * cos Φ ) = m * ( v_i * sin θ - v_f * sin Φ )

Note the definition of Impulse, I = m * Δv = Δp

Section A: Coefficient for Restitution

COR = v_fy / v_iy

v_fy = v_iy * COR

Hence: v_fy - v_iy = v_iy * COR - v_iy = v_iy * (COR - 1)

Section B: Angle After Bounce

tan Φ = v_fy / v_fx
tan Φ = (v_iy * COR) / v_fx
cot Φ = v_fx / (v_iy * COR)
COR * cot Φ = v_fx / v_iy

Section C: Angle Before Bounce

tan (-θ) = -v_iy / v_ix
- tan θ = - v_iy / v_ix
tan θ = v_iy / v_ix

Main:

Finding Final Angle After Bounce

Impulse, taking friction ( μ ) into consideration:

m * μ * (v_fy - v_iy) = m * (v_fx - v_ix)
μ * (v_fy - v_fi) = v_fx - v_ix

From Section A:

μ * v_iy * (COR - 1) = v_fx - v_ix
μ * (COR - 1) = v_fx/v_iy - v_ix/v_iy

From Section B:

μ * (COR - 1) = COR * cot Φ - v_ix/v_iy

From Section C:

μ * (COR - 1) = COR * cot Φ - tan θ
μ * (COR - 1) + tan θ = COR * cot Φ
(μ * (COR - 1) + tan θ) / COR = cot Φ
COR / (μ * (COR - 1) + tan θ) = tan Φ

which implies that:

Φ = arctan ( COR / (μ * (COR - 1) + tan θ) )

Finding Final Velocity

From Section B:

v_fx = (v_iy * COR) / tan Φ

We also know that:

cos Φ = v_fx / v_f

Then:

v_f = v_fx / cos Φ
v_f = (v_iy * COR) / (tan Φ * cos Φ)
v_f = (v_iy * COR) / sin Φ
v_f = (v_i * sin θ * COR) / sin Φ

Results

Final Angle:

Φ = arctan ( COR / (μ * (COR - 1) + tan θ) )

Final Velcoity:

v_f = (v_i * sin θ * COR) / sin Φ

Sources:

"Friction" http://physics.info/friction Retrieved 3/25/2012

"Coefficient of Restitution" Wikipedia. http://en.wikipedia.org/wiki/Coefficient_of_restitution Retrieved 3/22/2012

H. Brody "That's how the ball bounces" from "The Physics of Sports" edited by Armenti Angelo. American Institute of Physics: New York. 1992

This blog is property of Edward Shore. © 2012