Physics 2211
Quiz #1
Solutions
Spring 2008
Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected.
I
. (16 points) A car is travelling in a straight line with velocity +
v
0
. At time
t
= 0, it begins to accelerate at
a
= +
k
√
t
, where
k
is a positive constant. Through what distance does the car travel between times
t
= 0
and
t
=
T
? Express this distance in terms of any or all of
v
0
,
k
,
T
, and physical or mathematical constants.
.
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The acceleration is the timederivative of the velocity, so
a
=
dv/dt
⇒
v
=
Z
a dt
=
Z
kt
1
/
2
dt
=
k
t
3
/
2
3
/
2
+
C
where
C
is an integration constant. At time
t
= 0, the velocity is
v
0
, so
v
0
=
2
3
k
‡
0
3
/
2
·
+
C
⇒
C
=
v
0
so
v
=
2
3
kt
3
/
2
+
v
0
Since the velocity is the timederivative of the position, the displacement will be the definite integral of the
velocity over time.
v
=
dx/dt
⇒
Δ
x
=
Z
t
2
t
1
v dt
=
Z
T
0
‡
2
3
kt
3
/
2
+
v
0
·
dt
=
2
3
k
t
5
/
2
5
/
2
+
v
0
t
fl
fl
fl
fl
T
0
So
Δ
x
=
4
15
k T
5
/
2
+
v
0
T
Quiz #1 Solutions Page 1 of 5
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II
. (16 points) You are motorcycling home, when you find your way blocked by a swollen stream.
At that
instant, you are directly north of your home, and a distance
D
away, as illustrated. You travel a distance
a
along the stream, in a direction
θ
south of east, to a bridge. How far from your home is the bridge, in terms
of any or all of
D
,
a
,
θ
, and physical or mathematical constants? (The bridge has negligible length, so it
does not matter which side you are on.)
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Let the vector
~
D
point from the starting position to home. Let
the vector
~a
point from the starting position to the bridge. Let
the vector
~
b
point from the bridge to home. Then
~a
+
~
b
=
~
D
.
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 Spring '08
 UZER
 Physics, Velocity, Solutions Page, Car A

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