# Elastic Collision: Calculate the Percent Change in KE

## Homework Statement

A 0.26 kg cue ball with a velocity 1.2 m/s collides elastically with a 0.15 kg billiard ball at rest.
What percentage of the initial kinetic energy is transferred to the billiard?

m1= 0.26kg
Vi1= 1.2 m/s
Vf1= ?
m2= 0.15kg
Vi2= 0 m/s
Vf2= ?

## Homework Equations

(m1v2+m2v2)i=(m1v2+m2v2)f

ΔKE= ((.5)(m1)(Vi1)2+(.5)(m2)(Vi2)2)i -((.5)(m1)(Vf1)2+(.5)(m2)(Vf2)2)f

%ΔKE= 100(F-I/I)

## The Attempt at a Solution

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(-vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(-vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(-vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (-0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(-0.15kg)=(vf2)

(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)[[(0.312 kg m/s)+(0.26kg)(vf1)]/(-0.15kg)]
(0.312 kg m/s)=(0.26kg)(-vf1)-[(0.312 kg m/s)]-[(0.26kg)(vf1)]
(0.624 kg m/s)=(0.26kg)(-vf1)-[(0.26kg)(vf1)]
(0.624 kg m/s)=(-0.52vf1)
vf1= -1.2 m/s

((0.312 kg m/s)+(0.26kg)(-1.2 m/s))/(-0.15kg)=(vf2)
((0.312 kg m/s)+(-0.312 kg m/s))/(-0.15kg)=(vf2)
0=vf2

ΔKE= ((.5)(0.26kg)(1.2 m/s)2+(.5)(0.15kg)(0 m/s)2)i -((.5)(0.26kg)(Vf1)2+(.5)(0.15kg)(Vf2)2)f

The answer is suppose to be 93%

Last edited:

Doc Al
Mentor
(m1v2+m2v2)i=(m1v2+m2v2)f
That's one of the equations you'll need. Since you have two unknowns, you'll need a second equation. Besides momentum, what else is conserved?

## The Attempt at a Solution

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(-vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(-vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(-vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (-0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(-0.15kg)=(vf2)

(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)[[(0.312 kg m/s)+(0.26kg)(vf1)]/(-0.15kg)]
(0.312 kg m/s)=(0.26kg)(-vf1)-[(0.312 kg m/s)]-[(0.26kg)(vf1)]
(0.624 kg m/s)=(0.26kg)(-vf1)-[(0.26kg)(vf1)]
(0.624 kg m/s)=(-0.52vf1)
vf1= -1.2 m/s
You must have made a mistake somewhere. There's no way to solve for either final speed without using a second equation.

That's one of the equations you'll need. Since you have two unknowns, you'll need a second equation. Besides momentum, what else is conserved?

You must have made a mistake somewhere. There's no way to solve for either final speed without using a second equation.

Oh so I need to use ΔKE equation.

((.5)(m1)(Vi1)2+(.5)(m2)(Vi2)2)i = ((.5)(m1)(Vf1)2+(.5)(m2)(Vf2)2)f

and solve for either Vf1 or Vf2 then plug it back into:

(m1v2+m2v2)i=(m1v2+m2v2)f

Find the other final velocity, and then:

ΔKE= ((.5)(m1)(Vi1)2+(.5)(m2)(Vi2)2)i -((.5)(m1)(Vf1)2+(.5)(m2)(Vf2)2)f

%ΔKE= 100(F-I/I)

Doc Al
Mentor
Oh so I need to use ΔKE equation.

((.5)(m1)(Vi1)2+(.5)(m2)(Vi2)2)i = ((.5)(m1)(Vf1)2+(.5)(m2)(Vf2)2)f

and solve for either Vf1 or Vf2 then plug it back into:

(m1v2+m2v2)i=(m1v2+m2v2)f

Find the other final velocity, and then:

ΔKE= ((.5)(m1)(Vi1)2+(.5)(m2)(Vi2)2)i -((.5)(m1)(Vf1)2+(.5)(m2)(Vf2)2)f

%ΔKE= 100(F-I/I)
Good. That should do it.

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg)=(vf2)

((.5)(0.26kg)(1.2m/s)2+(.5)(0.15kg)(0 m/s)2)i = ((.5)(0.26kg)(-Vf1)2+(.5)(0.15)((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg))2)f
(.1872 kg m/s)=((.5)(0.26kg)(-Vf1)2+(.5)(0.15)((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg))2)
(.1872 kg m/s)=(-0.13kg)(vf1)2+(.13kg)[.0974kg m/s)+(0.0676)(vf1)2]/(.0225)

Isn't there a simpler way? Lol

((.5)(0.26kg)(1.2 m/s)2+(.5)(0.15kg)(0)2)i = ((.5)(0.26kg)(vf1)2+(.5)(0.15kg)(Vf2)2)f
(0.1872 kg m/s)+0=(0.13kg)(-vf1)2+(0.075kg)(vf2)
(0.1872 kg m/s)+(0.13kg)(vf1)2=(0.075kg)(vf2)2
SQR[(0.1872 kg m/s)+(0.13kg)(vf1)2]/(0.075kg)=vf
(2.895 m/s)(vf1)=VF2

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)((2.895 m/s)(vf1))
(0.312 KG M/S)+(0.26kg)(vf1)=(0.15kg)((2.895 m/s)(vf1))
[(0.312 KG M/S)+(0.26kg)(vf1)]/(0.4343KG M/S)
idk..

Last edited:
vela
Staff Emeritus
Homework Helper
Yes, there is if the collision is one-dimensional, which it looks like you're assuming here. Check your textbook for a simpler equation you can use for one-dimensional elastic collisions, rather than the conservation-of-kinetic-energy equation. It's simpler because it doesn't have the squared terms, so the algebra is easier to handle.

Providing the equation would have been less effort on both our parts. Thank you for the help. :)

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg)=(vf2)

(2.08 m/s)+1.73(vf1)=(vf2)

(1.2 m/s)+vf1=vf2
(2.08 m/s)+1.73(vf1)=(1.2 m/s)(vf1)
(2.08 m/s)=0.69(vf1)
vf1= 3 m/s

(2.08 m/s)+1.73(3m/s)=(vf2)
vf2=7.27m/s
ΔKE= (-(.5)(0.26 kg)(3 m/s)2+(.5)(0.15kg)(7.27 m/s)2)f-((.5)(0.26 kg)(1.2m/s)2+(.5)(0.15kg)(0)2)i
=-1.17+3.96J-0.1872J
=2.60J

(100x2.60J)/-1.17J+3.96J= 93%

This better be right. GRRRR.

Last edited:
bobie
Gold Member
vf1= 3 m/s

This better be right. GRRRR.
I might be mistaken, but I have studied that fv final velocity can never be more than 2iv
if vi1 = 1.2 vf1 should not exceed 2.4, do you agree?

Yeah I guess I need practice. I am having a hard time finding a good example. I haven't had math in a while so my algebra isn't very strong.

vela
Staff Emeritus
Homework Helper
Providing the equation would have been less effort on both our parts. Thank you for the help. :)
I figured you could use practice finding stuff in your book.

I get ##v_{1f} = 0.322\text{ m/s}## and ##v_{2f} = 1.52\text{ m/s}##. Check your sign convention for ##v_{1f}##. You're not being consistent.

How could both of my final velocities be positive if they are traveling in two different directions after collision?

vela
Staff Emeritus
Homework Helper
How do you know they're traveling in different directions after the collision?

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg)=(vf2)

(2.08 m/s)+1.73(vf1)=(vf2)

(1.2 m/s)+vf1=vf2
(2.08 m/s)+1.73(vf1)=(1.2 m/s)(vf1)
(2.08 m/s)=0.69(vf1)
vf1= 3 m/s

(2.08 m/s)+1.73(3m/s)=(vf2)
vf2=7.27m/s
ΔKE= (-(.5)(0.26 kg)(3 m/s)2+(.5)(0.15kg)(7.27 m/s)2)f-((.5)(0.26 kg)(1.2m/s)2+(.5)(0.15kg)(0)2)i
=-1.17+3.96J-0.1872J
=2.60J

(100x2.60J)/-1.17J+3.96J= 93%

This better be right. GRRRR.

(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg)=(vf2)

(2.08 m/s)+1.73(vf1)=(vf2)

(v1-v2)i=(v2-v1)f
1.2-(vf1)=vf2
(1.2 m/s)-vf1=vf2

(2.08 m/s)+1.73(vf1)=(1.2 m/s)-(vf1)
(2.08 m/s)+2.73(vf1)=(1.2 m/s)
2.73(vf1)=-0.88 m/s
2.73(vf1)/(2.73 m/s)=-0.88 m/s/(2.73m/s)
vf1=-0.32 m/s)

(2.08 m/s)+1.73(-0.32m/s)=(vf2)
vf2=1.52 m/s

ΔKE= (-(.5)(0.26 kg)(-0.32 m/s)2+(.5)(0.15kg)(1.52 m/s)2)f-((.5)(0.26 kg)(1.2m/s)2+(.5)(0.15kg)(0)2)i
=-0.0133+.1732J-0.1872J
= ???

(100x2.60J)/-1.17J+3.96J= 93%

How do you know they're traveling in different directions after the collision?

How do I know that they are not? Lol

I guess I just assumed that it's not a glancing collision. I thought the two balls bounce off each other in opposite directions, so the vf1 would be negative. No angles were given.

Last edited:
(0.26kg)(1.2m/s)+0=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)=(0.26kg)(-vf1)+(0.15kg)(vf2)
(0.312 kg m/s)-(0.26kg)(-vf1)= (0.15kg)(vf2)
(0.312 kg m/s)+(0.26kg)(vf1)= (0.15kg)(vf2)
((0.312 kg m/s)+(0.26kg)(vf1))/(0.15kg)=(vf2)

(2.08 m/s)+1.73(vf1)=(vf2)

(v1-v2)i=(v2-v1)f
1.2-(vf1)=vf2
(1.2 m/s)-vf1=vf2

(2.08 m/s)+1.73(vf1)=(1.2 m/s)-(vf1)
(2.08 m/s)+2.73(vf1)=(1.2 m/s)
2.73(vf1)=-0.88 m/s
2.73(vf1)/(2.73 m/s)=-0.88 m/s/(2.73m/s)
vf1=-0.32 m/s)

(2.08 m/s)+1.73(-0.32m/s)=(vf2)
vf2=1.52 m/s

ΔKE= (-(.5)(0.26 kg)(-0.32 m/s)2+(.5)(0.15kg)(1.52 m/s)2)f-((.5)(0.26 kg)(1.2m/s)2+(.5)(0.15kg)(0)2)i
=-0.0133+.1732J-0.1872J
= ???

(100x2.60J)/-1.17J+3.96J= 93%

AHA.:rofl:

ΔKEcue= ((.5)(0.26 kg)(1.2m/s)2+(.5)(0.15kg)(0)2)i-((.5)(0.26 kg)(-0.32 m/s)2)f
=0.1872J-0.0133J
=.1739J

100(.1739J)/.1872= 93%

Finalllyyyyyyyy:!!)

vela
Staff Emeritus
Homework Helper
How do I know that they are not? Lol

I guess I just assumed that it's not a glancing collision. I thought the two balls bounce off each other in opposite directions, so the vf1 would be negative. No angles were given.
Think about what happens when a bowling ball collides elastically with a ping pong ball at rest. The bowling ball is going to keep moving in the same direction as it was before because it's so much more massive than the ping pong ball. A collision doesn't have to cause the direction of its velocity to change.

bobie
Gold Member
How could both of my final velocities be positive if they are traveling in two different directions after collision?
How do I know that they are not? Lol
.
If the masses are equal the moving mass stops: fv1 = 0,
if the moving mass is smaller it rebounds with max fv1 =≈ 2 iv1
if it is greater it continues in the same direction with max fv1 = ≈ 1 iv1

93.1% is not correct,
you need not find fv1: Ef2/Ei1:
the exact result is 92.8019036287 % (0.34745/0.3744) *100

Last edited:
bobie
Gold Member
Isn't there a simpler way? Lol
Cheer up Arizona, when you get more practice you you can find the result in a simple way: 4Mm/(M+m)2:$$\Delta E = \frac{4*.26*.15}{(.26+.15)^2}$$

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