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The coefficient of restitution of a ball measures how "Lively" the bounce is. By definition, the coefficient equals $\frac{v_{2}}{v_{1}},$ where $v_{1}$ is the (downward) speed of the ball when it hits the ground and $v_{2}$ is the (upward) launch speed after it hits the ground. If a ball is dropped from a height of $H$ feet and rebounds to a height of $c H$ for some constant $c$ with $0<c<1$, compute its coefficient of restitution.

$h(t)=16 t^{2}, \sqrt{c}$

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

Applications of the Definite Integral

Section 5

Projectile Motion

Integrals

Integration

Applications of Integration

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

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06:26

Amplified Rebound Height T…

I'm gonna be dropping a ball here from some high age and I'm gonna let it bounce up and we're gonna be told that it bounces to some height. That's a little bit less than that. Call CH. The flexibility of this ball is equal to a constant C, which is V two over V one. The two ist of loss of the ball leaving the table Kiwanis the velocity of the ball when it hits the table first for the velocity of the ball hitting the table, the position function of the ball on the downward flight His negative 16 t squared No Visa bot, since it had no initial velocity. Plus, it's starting height. The velocity of the ball is the derivative of that which would be negative. 32 t Now, in order to figure the velocity when it hits the table, let's find when that position is zero. So fill in zero for why one of t that zero equals negative. 16 t squared plus age to track the age divine by the negative 16 which would give me t squared over a squared equals h over 16. And if I take the square root of ant T is the swear it of age over four. So that's the time it took the fall for its velocity. I'm filling in that time. And remember, that just means I'm gonna fill in the squared of H over four for T. So that's negative. 32 times the square root of H over four, which would be negative. Eight, I swear it. Eight. So that is my velocity wanted him off the table live you want for my V two. I'm gonna look at this a little bit differently. We had it bouncing upward to some height of C H. But if you reverse that, if you dropped an object from a height of CH, the speed that it hits the table at is equal to the speed of a bouncing ball that goes up to CH when it leaves the table. So this is gonna look similar to the first part. I'm going to say now that in the second ball, which I'm saying is dropped from a height of ch that its position function his negative 16 t squared plus ch. The derivative of that is its velocity, which would be negative 32 t and the CH derivative out there, A bit of a constant zero. Okay, so when does this at the table? Let's let why some two t equals zero again. The spindle it very similar to the previous problem. I will subtract ch I will divide by the negative 16. So now I have t squared equals C H over 16. And taking the square root gives me a time of the square root of ch over fear. Now for its velocity at that time and gonna look at the to prime, I will fill in that time. And if I feel that in for T, I'm gonna have negative 32 times the square root of C H over four or negative eight. Swear roots. See age now, technically, that was for the speed of it going downward for me, the velocity of the going downward, the velocity going upward would be the positive of that. Remember, we are concerned about speeds here, not velocity. So it's really not a problem that we have a negative. There were just gonna turn it to positive because it's a speed now. The flexibility value is V sub to over v sub one. Our velocity of it. Leaving the table was negative. Eight square root ch Our velocity of it Hitting the table was h square root age. Well, im sorry and get rid of the neck. If I just said that the negative is unimportant because this is speed, so we don't need a direction. The H will cancel and using your expert for your square rules. You can also cancel the ages so we end up in this case of a bounce to a height of C H. The flexibility constant is the square root of see.

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