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