Find Coefficient of Restitution with just height of the ball after one bounce

[x2017]

Homework Statement

A ball is dropped from a height of 9.02 m. After hitting the ground, the ball then rebounds to a height of 1.78 m. What is the coefficient of restitution associated with the ball and ground impact?

Homework Equations

e=(V'-v')/(V-v)

The Attempt at a Solution

I thought maybe this was a trick question and you just do 9.02-1.78=7.24 and that was the answer but I am wrong... I need a bit of guidance on how to proceed with this question.

EDIT:
I just realized I should be using e=v'/v instead since there is only one ball... And I think I have realized that v' is -9.81? Not completely sure though because I also think that that might only be acceleration... CONFUSED lol
I thought that maybe I could use acceleration to find velocity, but I don't have the time...

 

[haruspex]

And I think I have realized that v' is -9.81? Not completely sure though because I also think that that might only be acceleration...

Yes, that's the acceleration. What SUVAT equation do you know that relates acceleration, distance and speeds?

 

[x2017]

Yes, that's the acceleration. What SUVAT equation do you know that relates acceleration, distance and speeds?

v_f²=v_i²+2ad perhaps?
Thanks for the reply, I always seem to forget about the SUVAT equations...

v_f²=v_i²+2ad
v_f²-v_i²=2(9.81)(9.02)

I used 9.02m and not 1.78m because it traveled down 9.02m when gravity was acting on it and I don't know what it's acceleration upwards is. Hopefully this is correct!

v_f²-v_i²=2(9.81)(9.02)
v_f²-v_i²=176.9724
v_f-v_i=13.30
Δv=13.30m/s

Is the above something I can do? So now I have the change in velocity of the ball...
But only on the way down correct? or is this all together?

 

[haruspex]

v_f²=v_i²+2ad perhaps?
Thanks for the reply, I always seem to forget about the SUVAT equations...

v_f²=v_i²+2ad
v_f²-v_i²=2(9.81)(9.02)

I used 9.02m and not 1.78m because it traveled down 9.02m when gravity was acting on it and I don't know what it's acceleration upwards is. Hopefully this is correct!

v_f²-v_i²=2(9.81)(9.02)
v_f²-v_i²=176.9724
v_f-v_i=13.30
Δv=13.30m/s

Is the above something I can do? So now I have the change in velocity of the ball...
But only on the way down correct? or is this all together?

You have found the landing speed, yes. A few doubtful steps along the way, though.
You should be careful with signs. If up is positive, the acceleration is -9.81m/s², and the distance is -9.02m. That way you get a positive Δ(v²).
Secondly, you cannot go straight from v_f²-v_i² to v_f-v_i by square rooting. You need to plug in the value for v_i (0) first.

Now you need to find the rebound speed by the same method.

 

[x2017]

You have found the landing speed, yes. A few doubtful steps along the way, though.
You should be careful with signs. If up is positive, the acceleration is -9.81m/s², and the distance is -9.02m. That way you get a positive Δ(v²).
Secondly, you cannot go straight from v_f²-v_i² to v_f-v_i by square rooting. You need to plug in the value for v_i (0) first.

Now you need to find the rebound speed by the same method.

How do I get the acceleration for the rebound speed though? Since it's traveling upwards it's not gravity right? And I don't have enough information to solve for acceleration with Δv/Δt.

 

[haruspex]

How do I get the acceleration for the rebound speed though? Since it's traveling upwards it's not gravity right?

why not? Does gravity stop acting on a body just because it is moving upwards?

 

[x2017]

why not? Does gravity stop acting on a body just because it is moving upwards?

No, I guess gravity is causing it to slow down as it approaches its peak height! Okay I understand that part now, thanks!

I solved for the rebound speed and got 5.91m/s.

e=v'/v
e=5.91/13.30
e=0.44 which comes up as correct!

This turned out to be pretty simple, thanks for helping me understand how to go about it haruspex! :)

 

[haruspex]

No, I guess gravity is causing it to slow down as it approaches its peak height! Okay I understand that part now, thanks!

I solved for the rebound speed and got 5.91m/s.

e=v'/v
e=5.91/13.30
e=0.44 which comes up as correct!

This turned out to be pretty simple, thanks for helping me understand how to go about it haruspex! :)

Well done.

By the way, if you work it symbolically instead of plugging in numbers you will find the relationship between e and the height ratio. You don't actually need to calculate the velocities.

 

[haruspex]

What is that relationship between e and the height ratio?

How about you try what I suggested, solving the problem but entirely symbolically, i.e. not plugging in any numerical values.

 

[haruspex]

I only need the equation for some extra credit for a class that's not even relevant to my major. But thanks for the sarcasm.

There was no sarcasm, none at all. It was a serious suggestion for what would be far more useful to you than my merely providing an equation.
Indeed, that is how these forums work - we provide hints, corrections, advice, but require the student to show effort. You had effectively asked me to violate forum rules.