When the ball is initially dropped, its motion is governed by the position
function for objects in free fall:
(1)
x
1(t) = - 
gt
2 + h
When it bounces back up, it behaves like the example of a bullet
being fired upwards, and hence obeys the equation:
(2)
x
2(t) = -
gt
2 + v
0
t
where
v
0 = - .8v
, and
v
is the velocity at which the ball initially hits
the ground. To solve this problem, we must first use (1) to determine the
velocity at which the ball hits the ground. Then we can plug this value
into (2) and solve for the maximum height reached by the ball after it has
bounced.
From (1) we can find two important pieces of information. First of all,
v
1(t) = x
1'(t) = - gt
. Secondly, setting
x
1(t) = 0
we discover that the
ball hits the ground at time
t
f = 
. Plugging this value
into the velocity equation, we find that the velocity of the ball as it hits
the ground is
v = v
1(t
f) = - 
. Hence,
v
0 = - .8
v = .8
.
Now we can move onto equation (2). We won't plug in the value for
v
0
just yet, though. First we need to figure out at what point after the ball
has bounced it will reach its maximum height. The following observation
proves valuable: the ball's velocity is zero when it reaches its maximum
height. To use this fact, we need to find
v
2(t)
and set this
equation equal to zero in order to solve for the appropriate time.
v
2(t) = x
2'(t) = - gt + v
0
, so solving for
v
2(t) = 0
yields
t
max = 
, the time at which the ball reaches its maximum
height after the bounce. Plugging this time back into the equation for
x
2(t)
yields the actual height the ball attains at this time:
x
2(
t
max) =
Now, plugging in the value we found previously for
v
0
, we get our final
answer:
h
max = x
2(t
max) = .64h (Notice that .64 = (.8)2!)
There's actually a much easier way to solve this problem, but it requires
using physics principles that don't get introduced as early as kinematics.
However, even though it requires quite a bit of effort, it is still quite
remarkable that we can solve this problem with only our knowledge of
one-dimensional kinematics!
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