# Ind the distance traveled by the ball

• corpuzdawn
In summary, the ball of mass m falls a distance h[0] above a horizontal ceramic surface and bounces purely vertically on one spot. If the coefficient of restitution between the ball and surface halves after every bounce, the distance traveled by the ball when it hits the surface for the third time is h[0]/4. The coefficient of restitution results in the height of the ball being halved after every bounce, and the kinetic energy and height are both quartered. The coefficient of restitution cannot change on each bounce, and the KE is conserved and becomes mgh at the top.
corpuzdawn
A ceramic ball of mass m falls form rest a distance h[0] above a horizontal ceramic surface. The subsequent motion of the ball is purely vertical, bouncing ONLY on one spot on the surface. If the coefficient of restitution between the ball and surface halves after every bounce, find the distance traveled by the ball when the ball hits the surface for the third time.

is the distance traveled 0? because the ball only bounces at one spot?

corpuzdawn said:
is the distance traveled 0? because the ball only bounces at one spot?

NO, they want you to find the vertical distance traveled by the ball.

corpuzdawn said:
If the coefficient of restitution between the ball and surface halves after every bounce.
My guess is this should be "the coefficient of restitution results in the height of the ball being halved after every bounce". The coeffiecient is constant, but the height of the ball is halved on each bounce. If the coefficient is halved, then you'd need to know the initial coefficient of restitution on the first bounce.

rcgldr said:
My guess is this should be "the coefficient of restitution results in the height of the ball being halved after every bounce". The coeffiecient is constant, but the height of the ball is halved on each bounce. If the coefficient is halved, then you'd need to know the initial coefficient of restitution on the first bounce.

Take care here.
If the coefficient of restitution is 0.5 then the velocity will be halved. What does that say about the Kinetic Energy remaining and what does that say about the Height of the subsequent bounce?

sophiecentaur said:
Take care here.
If the coefficient of restitution is 0.5 then the velocity will be halved. What does that say about the Kinetic Energy remaining and what does that say about the Height of the subsequent bounce?

if the velocity is halved.. the KE is also halved... which means... the height is quartered?

What is the formula for ke?

oops. sorry. KE= [1/2]mv^2 so... the KE is quartered?

That's better. So height is a quarter, too. Mgh = KE

so that's the answer? the distance traveled is... h[0]/4

Yep.
That 50% gives an over-optimistic view of what's really going on. In fact, 3/4 of the original energy's lost.

rcgldr said:
My guess is this should be "the coefficient of restitution results in the height of the ball being halved after every bounce". The coeffiecient is constant, but the height of the ball is halved on each bounce. If the coefficient is halved, then you'd need to know the initial coefficient of restitution on the first bounce.

sophiecentaur said:
If the coefficient of restitution is 0.5 then the velocity will be halved. What does that say about the Kinetic Energy remaining and what does that say about the Height of the subsequent bounce?
Sorry for my somwhat confusing response, Cr (coefficient of restitution) = ratio of (speed after bounce) / (speed before bounce) or sqrt( (height after bounce) / (height before bounce) ).

The issue I was trying to point out is tha the OP states that Cr is halved on each bounce, versus stating that the speed or the height was being halved on each bounce. My guess is that the original problem meant that the speed (Cr = .5) or the height (Cr = sqrt(.5) ~= .7071) was halved on each bounce, not the Cr itself.

Last edited:
Absolutely. The COR can hardly change on each bounce.

sophiecentaur said:
That's better. So height is a quarter, too. Mgh = KE

why is KE=mgh? is it PE=mgh?

mass X g X height ? Not familiar?

btw, if they had wanted the aswer for "distance traveled" to be zero, they would probably have said "displacement" which is a vector quantity. Distance is a scalar.

sorry... but I always thought that mgh is PE. why is KE=mgh again?

Oh I see what you mean. The KE becomes mgh (at the top) was what I meant. (conservation)

## 1. How can I find the distance traveled by the ball?

To find the distance traveled by a ball, you will need to know the initial velocity, acceleration, and time. You can use the formula d = v0t + 1/2at2 to calculate the distance traveled.

## 2. Is there an easier way to calculate the distance traveled by a ball?

Yes, you can also use the formula d = (v0 + v)t/2, where v0 is the initial velocity, v is the final velocity, and t is the time. This formula is useful when the final velocity is known.

## 3. Can I calculate the distance traveled by a ball without knowing the initial velocity?

Yes, if the ball is thrown vertically upwards, you can use the formula d = 1/2gt2, where g is the acceleration due to gravity (9.8 m/s2) and t is the time the ball stays in the air.

## 4. How does air resistance affect the distance traveled by a ball?

Air resistance can significantly affect the distance traveled by a ball. As the ball moves through the air, it experiences a force in the opposite direction of its motion, which can slow it down and decrease the distance it travels.

## 5. Can the distance traveled by a ball be negative?

No, the distance traveled by a ball cannot be negative. Distance is a scalar quantity, meaning it only has magnitude and no direction. It is always measured as a positive value.

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