CK-12 Physics Concepts - Intermediate
Answer Key
Chapter 5: Motion in Two Dimensions
5.1 Projectile Motion for an Object Launched Horizontally
Practice
Questions
1. What was the problem the Mythbusters had with the dropped bullet? Why was
fixing this so important?
2. Why did they move the bullet being dropped to 360 ft away?
3. What was the final result?
4. What are some possible reasons that the bullets didn’t hit the ground at exactly
the same time?
Answers
1. They were having a hard time getting the falling bullet to release at the
exact moment the gun fired. It was important to fix this because they were
trying to test that the two bullets still hit the ground at the same moment.
2. They moved the bullet being dropped so that it would be at about the
same location that the bullet being fired was going to land.
3. They determined that the two bullets hit the ground about 39 milliseconds
apart, which is faster than humans can blink or even see at all.
4. The bullet being shot may have been shot at a slight (though very small)
angle upwards; this would increase the amount of time before it landed.
The two bullets were traveling through the air in different orientations; they
may have experienced different amounts of air resistance. Though the test
with the two bullets next to each other had them dropping within one
millisecond, it is possible that changing the location slightly changed the
relationship of their drops. Many other reasons exist, but the important
thing to remember is that the results are still significant.
Review
Questions
1. If a bullet is fired from a high powered rifle at the exact time a duplicate bullet is
dropped by hand near the barrel of the rifle, which bullet will hit the ground first?
a. the one dropped straight down
b. the one fired horizontally
c. both will hit the ground at the same time
1
2. A cannon is fired from the edge of a small cliff. The height of the cliff is 80.0
m. The cannon ball is fired with a perfectly horizontal velocity of 80.0 m/s. How
far will the cannon ball fly horizontally before it strikes the ground?
3. A cliff diver running 3.60 m/s dives out horizontally from the edge of a vertical cliff
and reaches the water below 2.00 s later. How high is the cliff and how far from
the base of the cliff did the diver hit the water?
Answers
1. c. Both hit the ground at the same time. For proof, see the practice video.
2. The cannon ball will fly 320 m before hitting the ground.
1
2d
2(80m)
d  at 2 t 

 4s
2
a
9.8m /s2
d
v  d  vt  (80m /s)(4s)  320m
t
Note that in these two equations, the d represents different values. The d in the
first equation is often written as h; it is the height of the cliff. The d in the second
equation is sometimes seen as x; it is the distance the cannon call goes before
hitting the ground.
1
1
d  h  at 2  (9.8m /s2 )(2.00s) 2  9.8m
2
2
3.
d
v  vt  d  x  (3.60m /s)(2.00s)  7.20m
t
The cliff is 9.8 m high, and the cliff diver gets 7.2 m away from the cliff
before he hits the water.


5.2 Projectile Motion for an Object Launched at an Angle
Practice
Questions
1. What is the cannon ball in this video?
2. What is used as the monkey in this video?
3. Does the velocity of the cannon ball matter, or will it hit the monkey at any velocity?
Answers
1. The cannon ball is actually a golf ball.
2. The monkey is a small stuffed monkey with a magnet in his head to hold him up.
3. The velocity of the cannon ball does not essentially matter. As long as the
cannonball is fired with enough force to reach the stand, it will hit the monkey.
As the monkey falls due to gravity, so also does the cannonball.
2
Review
Questions
1. A player kicks a football from ground level with a velocity of magnitude 27.0 m/s at
an angle of 30.0° above the horizontal.
a. Find the time the ball is in the air.
b. Find the maximum height of the ball.
c. Find the horizontal distance the ball travels.
2. A person standing on top of a 30.0 m high building throws a ball with an initial
velocity of 20. m/s at an angle of 20.0° below horizontal. How far from the base of
the building will the ball land?
3. An arrow is fired downward at an angle of 45 degrees from the top of a 200 m cliff
with a velocity of 60.0 m/s.
a. How long will it take the arrow to hit the ground?
b. How far from the base of the cliff will the arrow land?
Answers
1. To solve this problem, first separate the football’s trajectory into components:
vertical
sin(30) 
vertical  13.5m /s
27m /s
horizontal
cos(30) 
horizontal  23.4m /s
27m /s
v f  v i  at
v f  0m /s

a. v i  13.5m /s
a  gravity  9.8m /s2
v f  v i 13.5m /s
t

 1.38s
a
9.8m /s2
1
b. 𝑑 = 𝑣𝑖 𝑡 + 2 𝑎𝑡 2

𝑑 = (13.5
𝑚
1
𝑚
) (1.38𝑠) + (−9.8 2 ) (1.38) = 11.87
𝑠
2
𝑠
𝑑 = 11.87𝑚
1
d  v i t  at 2  v i t  0
c.
2
d  (23.4m /s)(1.38s)  32.3m

3
2. The ball will land 29.3 m away from the building.
3.
a. The arrow will take 3 seconds to hit the ground.
b. The arrow will land 127 meters from the cliff.
5.3 Circular Motion
Practice
Questions
1. What does centripetal mean?
2. What is uniform circular motion?
3. Why is centripetal acceleration always towards the center?
Answers
1. Centripetal means, “center seeking.”
2. Uniform circular motion is motion of an object at a constant speed in a circular
path.
3. To find centripetal acceleration, we add vf and the opposite of vi; the direction of
the resultant vector will always point towards the center.
Review
Questions
1. An automobile rounds a curve of radius 50.0 m on a flat road at a speed of 14 m/s.
What centripetal acceleration is necessary to keep the car on the curve?
2. An object is swung in a horizontal circle on a length of string that is 0.93 m long. If
the object goes around once in 1.18 s, what is the centripetal acceleration?
Answers
1. The centripetal acceleration is
2. The centripetal acceleration is
𝑣2
=
142
= 3.92 m/s2.
𝑟
50
4𝜋 2 𝑟
4(9.87)(.93)
𝑇2
=
1.182
5.4 Centripetal Force
Practice
Questions
4
= 26.36 m/s2.
1. Does the roller coaster in the video have a complete circle as part of its path?
2. What is it that keeps the glass of water on the tray as it swings over the student's
head?
3. What causes the centripetal force acting on the water in the cup? The roller coaster?
Answers
1. No.
2. The centripetal force keeps the water on the tray.
3. The string causes the centripetal force on the water. The track design does the
same in a roller coaster.
Review
Questions
1. A runner weighing 70 kg, moving at a speed of 8.8 m/s rounds a bend with a radius
of 25 m. What is the centripetal force needed to keep this runner on the curve and
what supplies this force?
2. A 1000. kg car rounds a curve of 50.0 m radius on a flat road with a speed of 14.0
m/s.
a. Will the car make the turn successfully if the pavement is dry and the coefficient
of friction is 0.60?
b. Will the car make the turn successfully if the pavement is wet and the
coefficient of friction is 0.20?
3. A 0.500 kg object tied to a string is swung around a person’s head in a horizontal
circle. The length of the string is 1.00 m and the maximum force the string can
withstand without breaking is 25.0 N. What is the maximum speed the object may
be swung without breaking the string?
Answers
𝑚𝑣 2
70𝑘𝑔∗8.82
𝑘𝑔∗𝑚
1. Using 𝐹 = 𝑟 =
= 216.8 𝑠2 = 217𝑁 The road supplies this force to the
25
runner.
2. To solve this problem, we have to determine both the centripetal force and the
frictional force.
mv 2
The centripetal force needed to make the curve is: F =
= 3920N
r
The friction depends on the normal force and the coefficient of friction. If the
frictional force exceeds the centripetal force, the car will make the turn.
a. Ff = mFN = (0.6)(9800N) = 5880N Yes; the car will make it around the turn.
b. Ff = mFN = (0.2)(9800N) =1960N No; the car will not make it around the turn.
5
mv 2
Fr
(25N)(1m)
3. F =
®v =
=
= 7.07m /s
r
m
0.5kg
5.5 Simple Harmonic Motion
Practice
Questions
1. What is the graph produced by a swinging pendulum's motion graphed over time?
2. How does the Exploratorium demonstrate the relationship between simple harmonic
motion and circular motion? Is it convincing?
3. Why don't the pendulums all swing at the same rate?
Answers
1. The graph produced by a swinging pendulum is a sine (or cosine) graph.
2. The Exploratorium has a setup with a rotating disk (circular motion) and a
swinging pendulum. The disk has a stick, the motion of which is projected onto
the same wall that the pendulum swings in front of. If you are careful, you can set
the pendulum to swing with the same rhythm as the disk’s rotation. The
pendulum is slowed over time by air resistance and friction, but the
demonstration clearly shows they follow the same pattern.
3. The pendulums have different lengths, which is why they don’t swing at the same
rate.
The following website has a set of questions and answers about simple harmonic
motion: http://www.education.com/study-help/article/simple-harmonic-motion_answer/
Review
Questions
1. In simple harmonic motion, when the speed of the object is maximum, the
acceleration is zero.
a. True
b. False
2. In SHM, maximum displacement of the mass means maximum acceleration.
a. True
b. False
3. If a spring has a spring constant of 1.00 × 103N/m, what is the restoring force when
the mass has been displaced 20.0 cm?
Answers
1. True
2. True
6
F = kx
3.
k = 1.00 ´ 10 3 N /m
x = 20.0cm = 0.200m
F = 200.N
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