PHYS 211-A
Exam I
Spring 2015
Name:___________________________
Instructions: Answer the multiple choice questions on this sheet by circling your answer. Only one answer per question.
Answer the partial credit problems in your green book. Start each numerical problem on a new page. Show all work for
full credit. Correct answers with little supporting information will receive very little credit. Make sure your name is on
both this sheet and the green test booklet. When you finish your exam, place this sheet inside your green test book and
turn both in at the front desk. You may keep (or discard) the equation sheet.
Multiple Choice: (4 points each)
For questions 1-3, use the picture shown below which illustrates an object undergoing projectile motion. The object
moves from left to right and the points A-E indicate its position at five different points along its trajectory. Allpoints
indicated are while the object is in the air and air resistance can be neglected.
1. Which one of the following, if any, correctly ranks the points of the trajectory according to speed from least to
greatest?
a. A B C D E
b. A B D E C
c. E D A B C
d. A E B D C
e. E D C B A
f. C D B E A
g. None of these
2. Which one of the following if any correctly lists the point or points of the trajectory where the y-component of
the velocity is negative and the y-component of the acceleration is negative?
a. A and B
B. A, B, and C c. C
d. D and E
e. D, C, and E
f. B and D
g. All of the points h. None of the points
3. Which one of the following, if any, correctly lists the point or points of the trajectory where the y-component of
the acceleration is negative and the x-component of the velocity is zero?
a. A and B
b. D and E
c. C
d. A B and C
e. C D and E
f. All of the points
g. None of the points.
4. Two cars are initially moving with speeds 𝑣𝐴 and 𝑣𝐵 . The cars are decelerated at the same rate until they come
to a stop. If it takes car A four times as far to stop as car B, then how do their initial speeds compare?
a. 𝑣𝑎 = 4𝑣𝑏
b. 𝒗𝒂 = 𝟐𝒗𝒃
c. 𝑣𝑎 = √2 𝑣𝑏
d. 𝑣𝑎 = 𝑣𝑏
1
e. 𝑣𝑎 = 2 𝑣𝑏
f. None of these
5. Which one of the following statements, if any, is not true about the cross product of two vectors?
a. 𝑖̂ × (𝑗̂ × 𝑘̂ ) = 0
⃗⃗ = −𝐵
⃗⃗ × 𝐴⃗
b. 𝐴⃗ × 𝐵
⃗⃗| is a maximum when the vectors are perpendicular.
c. |𝐴⃗ × 𝐵
⃗⃗ is a parallel to both 𝐴⃗ and 𝐵
⃗⃗.
d. 𝐴⃗ × 𝐵
e. None of the above.
Partial Credit:
6. A car accelerates from rest at 4.28 m/s2 for 4.54 s. The driver then immediately puts on the brakes and brings
the car to a stop in a distance of 25.6 m. Assume the car is moving in the positive x-direction.
a. What is the maximum speed the car reaches? (5 points)
m
m
m
𝑚
𝑣𝑚𝑎𝑥 = 𝑣0 + 𝑎1 𝑡1 = 0 + (4.28 2 ) (4.54 𝑠) = 19.43 ≈ 19.4
s
s
s
𝑠
b. What is the acceleration of the car as it comes to a stop? (10 points)
𝑣 2 = 𝑣02 + 2𝑎2 Δ𝑥.
𝑚 2
𝑚 2
𝑣 2 − 𝑣02 (0 𝑠 ) − (19.43 𝑠 )
𝒎
𝑎=
=
= −𝟕. 𝟑𝟕 𝟐 𝒊̂
2Δ𝑥
2(25.6 𝑚)
𝒔
c. What is the total distance traveled by the car from the time it begins speeding up until it comes to a stop?
(10 points)
While speeding up, the car travels
1
𝑚
𝑑1 = (4.28 2 ) (4.54 𝑠)2 = 44.1088 𝑚.
2
𝑠
The total distance traveled is then
𝑑 = 44.1 m + 25.6 m = 69.7 m
⃗
⃗⃗ =
7. You are given three vectors. Vector 𝐴 is 20.0 m at 30.0 degrees above the positive x-axis. Vector 𝐵
8.74 m 𝑖̂ − 12.7 m 𝑗̂. Vector 𝐶⃗ is 14.2 m at 237o from the positive x-axis.
Find the components of vectors 𝐴⃗ and 𝐶⃗. (5 points)
𝐴𝑥 = 𝐴 cos 𝜃𝐴 = (20.0 𝑚) cos 30𝑜 = 17.32 m
𝐴𝑦 = 𝐴 sin 30𝑜 = (20.0 𝑚) sin 300 = 10.0 m
𝐶𝑥 = 𝐶 cos 𝜃𝐶 = (14.2 𝑚) cos 237𝑜 = −7.734 m
𝐶𝑦 = 𝐶 sin 𝜃𝐶 = (14.2 𝑚) sin 237𝑜 = −11.909 m
⃗⃗. (5 points)
b. Find the magnitude and angle of vector 𝐵
a.
⃗⃗| = √𝐵𝑥2 + 𝐵𝑦2 = √(8.74 𝑚)2 + (−12.7 𝑚)2 = 15.4168 m
|𝐵
𝜃𝐵 = tan−1 (
−12.7
) = −55.5°
8.74
⃗⃗ × 𝐶⃗. (10 points)
c. Find 𝐵
⃗⃗ × 𝐶⃗ = |𝐵
⃗⃗||𝐶⃗| sin(𝜃𝐵 − 𝜃𝐶 ) (−𝑘̂ ) = (15.4168 𝑚)(14.2) sin(304.53° − 237°) (−𝑘̂ ) = 202 m2 (−𝑘̂)
𝐵
⃗⃗ − 𝐶⃗). (5 points)
d. Find (𝐴⃗ + 𝐵
⃗⃗ − 𝐶⃗ = (𝐴𝑥 + 𝐵𝑥 − 𝐶𝑥 )𝑖̂ + (𝐴𝑦 + 𝐵𝑦 − 𝐶𝑦 )𝑗̂
𝐴⃗ + 𝐵
⃗⃗ − 𝐶⃗ = (17.32 𝑚 + 8.74 𝑚 − (−7.734 𝑚))𝑖̂ + (10.0 𝑚 + (−12.7 𝑚) − (−11.909 𝑚))
𝐴⃗ + 𝐵
⃗⃗ − 𝐶⃗ = 33.8 m 𝑖̂ + 9.2 m𝑗̂
𝐴⃗ + 𝐵
8. A one-dimensional, critically damped harmonic oscillator undergoes motion given by
𝑥 = 𝐴𝑡𝑒 −𝛾𝑡
m
where 𝐴 = 2.75 and 𝛾 = 0.125 s−1. (Hint: For this problem you will need to use the product rule from
s
calculus).
a. Determine the velocity as a function of time and find the velocity of the oscillators at t=4.00s. (8 points)
𝑑𝑥⃗
𝑣⃗ =
= 𝐴𝑒 −𝛾𝑡 − 𝛾𝐴𝑡𝑒 −𝛾𝑡 = 𝐴𝑒 −𝛾𝑡 (1 − 𝛾𝑡).
𝑑𝑡
𝑚
m
−1
𝑣⃗(𝑡 = 4.00 s) = (2.75 ) 𝑒 (0.125 𝑠 )(4 s) (1 − (0.125)(4.00s)) =
𝑠
s
m
𝑣⃗(𝑡 = 4𝑠) = 2.27
s
b. Determine the acceleration as a function of time and find the acceleration of the oscillator at t=4.00s.
(7 points)
𝑎⃗ =
𝑑𝑣⃗
= −𝛾𝐴𝑒 −𝛾𝑡 (1 − 𝛾𝑡) − 𝛾𝐴𝑒 −𝛾𝑡 = 𝐴𝑒 −𝛾𝑡 (𝛾 2 𝑡 − 2𝛾)
𝑑𝑡
𝑎⃗(𝑡 = 4𝑠) = (2.75
𝑚 (0.125 𝑠−1 )(4 s)
)𝑒
((0.125 𝑠 −1 )2 (4 s) − 2(0.125 𝑠 −1 ))
𝑠
𝑎⃗(𝑡 = 4 𝑠) = −0.850
9.
m
s2
A projectile is launched into the air from ground level with an initial speed of 32.8 m/sat an angle of 54.3o above
the horizontal. The ball lands at the same level from which it is launched. Ignore air resistance.
a. What is the maximum height the projectile reaches above the ground? (8 points)
𝑦𝑚𝑎𝑥 =
2
−𝑣0𝑦
2𝑎𝑦
2
𝑚
) sin 54.3°)
𝑠
= 36.2 m
𝑚
2 (−9.8 2 )
𝑠
− ((32.8
=
b. How long is the projectile in the air? ( 7 points)
𝑚
𝑣𝑓𝑦 − 𝑣0𝑦 −𝑣0𝑦 − 𝑣0𝑦 2 (26.6363 𝑠 )
𝑡=
=
=
= 5.44 s
𝑚
𝑎𝑦
𝑎𝑦
−9.8 2
𝑠