Quiz 4
Math 254
Thursday, April 23
Name:
Recitation:
Unjustified answers will receive no credit. Justification includes mathematical work as
well as verbal explanations. Verbal explanations should be complete sentences. For multiple
choice questions you get 1pt for the correct answer and up to 4pts for the explanation.
(5pts) Problem 1:
An uncompressed spring can be modeled by
~r(t) = h20 cos(2πt), 20 sin(2πt), 4ti
Which of the following gives length of metal used in such a spring as 0 ≤ t ≤ 10?
R 10 √
R1√
A. 0
C. 0 1616π 2 dt
1616 dt
R 10
D. 0 h−40π sin(2πt), 40π cos(2πt), 4i dt
R 10 √
1600π 2 + 16 dt
B. 0
E. None of these is correct.
Solution: Recall that the arc-length of a vector-parametrized curve is given by
Z b
||~r 0 (t)|| dt
s=
a
. Then
~r 0 (t) = h−40π sin(2πt), 40π cos(2πt), 4i
q
2
||~r 0 (t)|| =
− 40π sin(2πt) + 40π cos(2πt) + (4)2
q
= 1600π 2 (sin2 (2πt) + cos2 (2πt)) + 16
p
= 1600π 2 (1) + 16
And so B. is the correct answer.
(5pts) Problem 2:
Safeco Field (Seattle’s baseball stadium) is oriented so that the left field foul pole is
north of home plate and the right field foul pole is east of home plate. Suppose Alex
hits a baseball in Safeco Field directly northeast and the ball leaves the bat at a speed
of 46 m/s and at an angle of about π6 radians above the horizontal ground. A cross-wind
blows north producing an acceleration of 0.75 m/s2 to the north. You may assume the
ball is 1m off the ground when it is hit.
With no interference, where would the ball hit the ground?
Solution: With the given information,
~v0 = 46 cos π6 cos π4 , 46 cos
~r0 = h0, 0, 1i
~a(t) = h0, 0.75, −9.8i
π
6
sin
π
4
, 46 sin
π
6
We proceed by integrating the acceleration vector to get velocity, and then integrating
velocity to get position (incorporating the appropriate constants as we go):
~a(t) = h0, 0.75, −9.8i
Z
↓
D √
E
√
~v (t) = 23 26 , 0.75t + 23 26 , −9.8t + 23
Z
↓
D √
E
√
~r(t) = 23 26 t, 0.375t2 + 23 26 t, −4.9t2 + 23t
The ball will hit the ground when z(t) = −4.9t2 + 23t = 0, which happens when
t ≈ 4.694 seconds. Then ~r(4.694) ≈ h132.22, 140.48, 0i, so the baseball lands 132.22
meters East and 140.48 North of home plate.
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