# How do you find the coefficient of restitution with only d?

## Homework Statement

If a rubber ball is dropped from the height of 120 cm from the floor, and rebounds to a height of 80 cm, what is the value of e, the coefficient of restitution? Show your reasoning and justify your solution.

initial distance 1= 120 cm
final distance 1= 0 cm
initial distance 2 = 0 cm
final distance 2 = 80 cm
acceleration = gravity = -9.8 m/s^2
initial velocity 1 = 0 m/s
final velocity 2 = 0 m/s

## Homework Equations

Not very sure... I honestly don't know how to find the numerical answer here.

e = (vf1/vi1)

## The Attempt at a Solution

I can't come up with a number solution. I do have a justification, though. e will be less than 1 for sure because energy was lost during the rebound. You can tell energy was lost because the ball did not bounce back to original height. It's inelastic and not elastic, so e will not be equal to 1, and the energy is not fully conserved.

Now determining the value of e is where I'm stuck, unless that's impossible.

Thank you for any and all help!

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haruspex
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Gold Member
You know the acceleration and distance for both the fall and the rebound. You know the initial velocity of the one and the final velocity of the other. What two other velocities can you calculate?

You know the acceleration and distance for both the fall and the rebound. You know the initial velocity of the one and the final velocity of the other. What two other velocities can you calculate?
Oh! I think I just got a method!

Fall:
vf = sqrt(2(-9.8)(-1.20) = about 4.8 m/s

Rebound:

0 = (vi)^2 + 2(-9.8)(.80)
vi = sqrt(2(9.8)(.80)) = about 4.0 m/s

e = (Vf2 - vf1) / (vi2 - vi1) = (0 - 4.8) / (4.0 - 0) = -1.2

Although, I'm not sure if this is right. I checked it over and I don't really see anything wrong with it... I swear I remember that e is not allowed to leave the boundaries of 0 and 1.

haruspex
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Gold Member
e = (Vf2 - vf1) / (vi2 - vi1) = (0 - 4.8) / (4.0 - 0) = -1.2
Two mistakes there.
In the definition of e there is a sign reversal: (Vf2 - vf1) / (vi1 - vi2).
Secondly, the "vf" you found was the velocity at the end of the fall, but the end of the fall is the initial state for the bounce.

Two mistakes there.
In the definition of e there is a sign reversal: (Vf2 - vf1) / (vi1 - vi2).
Secondly, the "vf" you found was the velocity at the end of the fall, but the end of the fall is the initial state for the bounce.

So would vf1 be equal to vi2? But then wouldn't that be elastic instead of inelastic?

haruspex
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Gold Member
So would vf1 be equal to vi2? But then wouldn't that be elastic instead of inelastic?
There are three phases in all: the initial descent, the bounce, the final ascent.
Your original vf was the vf of the first phase, while your vi was the vi of the third phase.
How do these relate to the initial and final velocities of the bounce phase?

There are three phases in all: the initial descent, the bounce, the final ascent.
Your original vf was the vf of the first phase, while your vi was the vi of the third phase.
How do these relate to the initial and final velocities of the bounce phase?
I don't really know :/ my teacher is behind lessons in class, but he gives us lots of homework that is far in advance.

Does this have something to do with loss of kinetic energy?

haruspex
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Gold Member
I don't really know :/ my teacher is behind lessons in class, but he gives us lots of homework that is far in advance.

Does this have something to do with loss of kinetic energy?
It isn't a matter of knowing, it's a matter of thinking.
The "initial velocity" of the bounce Is the velocity immediately before impact. How does that relate to the initial and final velocities of the falling phase? (Don't overthink it, it is a very simple answer.)

It isn't a matter of knowing, it's a matter of thinking.
The "initial velocity" of the bounce Is the velocity immediately before impact. How does that relate to the initial and final velocities of the falling phase? (Don't overthink it, it is a very simple answer.)
Hm... so the initial velocity of the bounce is the final velocity of the fall, and the final velocity of the bounce is the initial velocity of the ascent? So the initial velocity of the fall and the final velocity of the ascent are still equal?

haruspex
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Gold Member
Hm... so the initial velocity of the bounce is the final velocity of the fall, and the final velocity of the bounce is the initial velocity of the ascent?
Yes.
So the initial velocity of the fall and the final velocity of the ascent are still equal?
Yes, those are both zero.

Yes.

Yes, those are both zero.
Took me an hour, but I think I got it!

The final velocity for fall is equal to the bounce's initial = 4.8 m/s
The initial velocity for ascent is equal to the bounce's final = 4.0 m/s

So really, the equation I did used the correct numbers, but everything was just misplaced!

e = (vf2 - final bounce velocity) / ((vi1 - initial bounce velocity) = -4.0 / -4.8 = approximately 0.83

Okay, so did I do it right now lol?

haruspex
Homework Helper
Gold Member
Took me an hour, but I think I got it!

The final velocity for fall is equal to the bounce's initial = 4.8 m/s
The initial velocity for ascent is equal to the bounce's final = 4.0 m/s

So really, the equation I did used the correct numbers, but everything was just misplaced!

e = (vf2 - final bounce velocity) / ((vi1 - initial bounce velocity) = -4.0 / -4.8 = approximately 0.83

Okay, so did I do it right now lol?
Yes, except that you have some accumulated errors there which make your answer a bit off.
This happens when you keep rounding intermediate numerical results.
You will get a more accurate answer if you keep everything algebraic until the final step. (There are many other advantages to that technique, and I very strongly encourage students to adopt it.) You will find that the answer is the square root of the ratios of the heights, giving about 0.816.

Yes, except that you have some accumulated errors there which make your answer a bit off.
This happens when you keep rounding intermediate numerical results.
You will get a more accurate answer if you keep everything algebraic until the final step. (There are many other advantages to that technique, and I very strongly encourage students to adopt it.) You will find that the answer is the square root of the ratios of the heights, giving about 0.816.
Ah okay! Thank you so much for the help! I really and truly appreciate it! Thanks a million! : )