Bouncing Ball

## Question:

Hi, I am a sophomore taking general physics one and was given the
following problem: "A ball falls off of a table and lands on
the floor below. It then bounces up .5 of the distance it fell
and repeats this process until it has bounced a total of three
times. If its last bounce took it to a height of 1 m, what was
the ball's total time in the air from its fall off the table
to the rise after its last bounce?" My question is,
"Can I assume that the initial velocity for each bounce is
equal to half of the final velocity from the proceeding impact,
in the opposite direction?"
## Answer:

To answer your question:

When a ball bounces (neglecting air resistance) it will rise to a
height where its potential energy equals the kinetic energy with
which it left the floor on the upward flight. The potential
energy is given by m*g*h where m is the mass of the ball, g is
the acceleration due to gravity and h is the height of the
bounce. The * symbol denotes multiplication.
In the situation you describe where the height of the ball is
reduced by half with each bounce, the energy of each bounce is
half the energy of the previous bounce, since the potential
energy is directly proportional to the height of the bounce, m
and g being constant. This means that the kinetic energy which is
given by 1/2*m*v^{2} is reduced by half. If we look at
the kinetic energy in one bounce compared to the kinetic energy
of the next bounce,

(1/2*m*v1^{2})/(1/2*m*v2^{2})=2

or,

v1^{2}/v2^{2}=2

or,

v1^{2}=2*v2^{2}

or,

v1=1.414*v2

rather than

v1=2*v2

as you suggest.

I would look at the problem this way:

A 1 meter height after the third bounce implies a 2 meter height
after the second bounce, a 4 meter height after the first bounce
and a 8 meter initial height. Use the relationship between
distance fallen and time of fall to calculate the time of the
initial 8 meter drop. Then, since a ball on the way up is subject
to exactly the same acceleration as a ball on its way down,
recognize that the rise time for each bounce will equal the fall
time. So calculate the fall time for a 4 meter fall and double
it, the fall time for a 2 meter fall and double that, and finally
the time for a 1 meter fall not doubled since we stop the problem
with the ball at the 1 meter height. Adding all these times
together should give you the answer.

Hope this helps.

J. D. Jones

M. Casco Associates