Ph1a: Solution to Quiz 1
Alejandro Jenkins, Fall 2004
Problem 1 (6 points)  Marble Descending Staircase
1
A marble bounces regularly down a long ﬂight of stairs, hitting each step with the same
speed and at the same distance from the edge, and then bouncing up to the same height
h
above the step, as shown in the ﬁgure.
l
l
h
v
v
i
f
Each stair has the same height and depth
`
. The
horizontal
component of the marble’s
velocity is constant throughout, but the stairs have the property that

v
f
/v
i
=
e
, where
v
i
and
v
f
are the
vertical
velocity components just before and just after the bounce respectively,
and
e
is a constant (0
< e <
1).
(a) (2 points)
Find an expression for
v
i
in terms of
e
,
`
, and the gravitational accelera
tion
g
.
Solution:
The marble bounces oﬀ the stair and begins moving upward with a
vertical velocity
v
f
=

ev
i
. After reaching the top of its bounce, it begins to
move downward. When it passes the level of the previous stair, it will have
vertical velocity
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 Fall '07
 Goodstein
 Derivative, Acceleration, Velocity, vertical velocity, gravitational acceleration, Car J

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