Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Crave the Wave, Feb 16, 2008
Mentor Invitational
Page 1 of 14
ANSWER KEY
Scoring: Points will be awarded for the accuracy and quality of the responses. Show equations
used to receive credit for a higher‐quality response. Ties will be broken using pre‐selected
questions.
1
The sound of a lightning flash is heard 5.0 seconds after the flash. Assume that the light
travels much more swiftly than the sound. Assume that the speed of sound is 330 m/s. How
far away was the lightning?
The sound took 5.0 seconds (since we assume that the light travels much more swiftly than
the sound, we can assume that the lightning flash reaches the observer instantly) to travel
from the flash to the observer. Use x = vt = (330 m/s)(5.0 s) = 1650 meters ! 1 mile. The
lightning is about 1 mile away for every 5 seconds.
2
a)
b)
a)
b)
3
The crowd at a football game enthusiastically creates a stadium wave. Each person stands up in turn
to create a ripple effect that flows around the stadium. A) Is this an example of a transverse wave or
a longitudinal wave? B) Why?
A stadium wave would be an example of a transverse wave because the particle/person moves
perpendicular to the direction of energy propagation.
4
Ocean waves smash into a breakwater at the rate of 12 per minute. The wavelength of these waves
is 39 m. What is their speed?
What is the frequency of a second hand on an analog clock?
What is the frequency of a minute hand on an analog clock?
The second hand makes a complete rotation in 60 seconds. It’s frequency is 1/60 Hz= 0.0167 Hz
The minute hand makes a complete rotation in 60 minutes. It’s frequency is 1/3600 Hz= 2.78 x
10‐4 Hz
#$
# %&'
f = "%&' )( () *+,) = 0.2 Hz
5
- . /0 . ".2) (39) = 7.8 m/s
Using the diagram below…
a)
b)
c)
d)
Which letter describes the amplitude of the wave?
Which letter represents the wavelength of the wave?
Label a wave crest as F
Label a wave trough as G
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 2 of 14
e) What is the frequency of this wave?
f) What is the period of this wave?
Using the diagram below…
a)
b)
c)
d)
e)
f)
6
Which letter describes the amplitude of the wave? D
Which letter represents the wavelength of the wave? A
Label a wave crest as F
Label a wave trough as G
Frequency = 2.5 Hz
Period = 0.4 seconds
In 12 days, you can listen to the first Indians spring training game on the Tribe’s Flagship
station WTAM 1100 which broadcasts at 1100 KHz. What is the wavelength of the radio
waves carrying the tribe games on this station? (Assume the speed of the radio waves is 3 x
108 m/s.
1
34 #)5 %/*
a) 0 . 2 = ##)) ))) 67= 272.7 m
7
The Figure below shows an incident wave pulse moving to the left.
The blue pulse encounters a mystery wave pulse from the right resulting in the superposition of the
waves as shown below in red.
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 3 of 14
Make a Sketch of the mystery wave pulse travelling to the right.
8
The middle C string of a piano vibrates with a frequency of 261.6 Hz when excited in its fundamental
mode. What are the frequencies of the first, second, and third harmonic modes of this string?
The fundamental mode for a string with fixed endpoints is a standing wave with
0# . 28. The first harmonic is a standing wave with 0$ . 8 . The general formula for the
'1
normal frequencies, or eigenfrequencies, of the string is /' . $9 . It is given in the problem
that 0# . 261.6 ;<. The first harmonic frequency is 0$ . 523.2 ;<. The 2nd harmonic
frequency is 03 . 784.8 ;<. The 3rd harmonic frequency is 0= . 1046.4 ;<.
9
a)
A transverse wave is transporting energy from east to west. The particles of the
medium will move in what direction?
b)
A longitudinal wave is transporting energy from north to south. The particles of the medium will
move in what direction?
North and south
North and south
a)
b)
10
A wave has amplitude of 7.5 cm and a frequency of 10 Hz, and the distance from a crest
to the nearest trough (measured along the direction of the wave’s energy transfer) is
measured to be 5 cm. Determine the period of such a wave.
T = 1/f = 0.1 seconds
Crave the Wave ANSWER KEY
11
Mentor Invitational Feb 16, 2008
Page 4 of 14
The drawing shows a 30-60-90 prism and 2 light rays, A and B, both of which strike the
prism perpendicularly. The prism is surrounded by an unknown liquid. When ray A
reaches the hypotenuse in the drawing, it is totally internally reflected. Which one of the
following statements applies to ray B when it reaches the hypotenuse? Why?
A) It may or may not be totally internally reflected, depending on what the
surrounding liquid is
B) It is not totally internally reflected, no matter what the surrounding liquid is
C) It is totally internally reflected, no matter what the surrounding liquid is.
The answer is C. If ray A is totally internally reflected, the critical angle must be less
than 30 degrees for these media. Therefore the 60 degree angle is guaranteed to also
be greater than the critical angle which will result in total internal reflection.
12 A car is travelling at a speed of 90 ft/s (61 mi/hr) along a road that runs parallel to a railroad
track. Practically straight in front of the car is a locomotive that is waiting for a repair. If the
speed of the sound is 1080 ft/sec and the driver of the car hears a frequency of 400 Hz when
the locomotive whistle is blown, what is the actual frequency of the whistle?
Use the Doppler‐effect equation:
2>
2@
= 1 A1
, where - 9 = velocity of listener, -* = velocity of source, - = speed of sound in
1?1
>
@
the medium. Since the source is not moving, -* = 0.
=))
2
=))"#)B)D
= @
/* = ##E)
= 369 Hz
#)B)?C) #)B)
13
An ocean wave has a wavelength of 120 m and a period of 8.77 s.
A) What is the frequency of the wave?
B) What is the speed of the wave?
a) f = 1/T = .114 Hz
b) - . /0 . ".114) (120) = 13.68 m/s
14
A loudspeaker produces a sound wave. Does the wavelength of the sound increase, decrease, or
remain the same, when the wave travels from the air into the water? Why?
The frequency of the wave remains the same. Sound travels faster in water than in air. The
relationship between frequency, wavelength, and speed is - . /0. In order for this to remain
true, the wavelength must also change if the speed of the wave changes and the frequency
remains constant.
15
The drawing below shows 3 layers of liquids, A, B, and C, each with a different index of refraction.
Light begins in liquid A, passes into B, and eventually into C, as the ray of light in the drawing shows.
The vertical lines denote the normals to the interfaces between the layers. Which liquid has the
smallest index of refraction? Why?
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 5 of 14
B. Liquid B has a smaller index of refraction than A because the light is bending away from the
Normal. B is smaller than C because the light is refracting toward the normal for the B/C interface.
16
The frequency of electromagnetic wave A is twice that of electromagnetic wave B. For these two
0
waves, what is the ratio 0F of the wavelengths in a vacuum?
G
A.
0F
0G
= 2, because wave A has twice the speed that wave B has.
B.
0F
0G
= 2, because wave A has one‐half the speed that wave B has.
C.
0F
0G
= $, because wave A has one‐half the speed that wave B has.
D.
0F
0G
= $, because wave A has twice the speed that wave B has.
E.
0F
0G
= $, because both waves have the same speed.
#
#
#
E.
17
The speed of a transverse wave on a string is 600 m/s, and the wavelength is 0.24 m. The amplitude
of the wave is 0.5 cm. How much time is required for a particle of the string to move through a total
distance of 1.0 km?
In 1 cycle, a particle will move 2 cm. Taking snapshots of a period for that particle, we’d see the
particle start at its zero‐position, then be displaced upward to be the particle at the crest which is 0.5
cm, then come back to zero which is 0.5 more cm, then displaced downward to the trough which is
another 0.5 cm, and finally 0.5 cm back up to zero to complete the period of motion. Using - . /0
and substituting the given speed of the wave and the wavelength, we can find that the
frequency is 2500 Hz. We want to know how much time it will take to move 1 km by 2 cm
increments to find out how many cycles it will take for the particle to travel 1 km.
# 4 #)H
$ 4 #)IJ
=
50,000 cycles. Divide how many cycles it will take by the frequency to get the number of
seconds it will take.
18
K),)))
,L,M+*
$K)) ,L,M+*/*+,N'O
= 20 seconds.
The drawing shows three situations A, B, and C in which an observer and a source of electromagnetic
waves are moving along the same line. In each case the source emits a wave of the same frequency.
The arrows in each situation denote velocity vectors relative to the ground and have magnitudes as
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 6 of 14
shown of either v or 2v. Rank the frequencies of the observed waves in descending order (largest
first) according to magnitude.
B, A, C
19
For each part of the Electromagnetic Spectrum listed below, indicate the order of the spectrum
where 1 is the part of the spectrum with the lowest quantum energy and 10 is the part of the
spectrum with the highest quantum energy.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
20
! Infrared
! Microwaves, Radar
! Short wave radio
! Visible Light
! TV and FM Radio band
! X‐rays
! AM radio
! Gamma rays
! Ultraviolet Light
! Millimeter waves, telemetry
6 Infrared
4 Microwaves, Radar
2 Short wave radio
7 Visible Light
3 TV and FM Radio band
9 X‐rays
1 AM radio
10 Gamma rays
8 Ultraviolet Light
5 Millimeter waves, telemetry
Suppose that a pair of flutes generates sounds waves of frequency 264 Hz and 262 Hz
respectively. What is the beat frequency?
/P+QR . |/Q S /P | = 2 Hz, hence a listener will feel a beat pulse 2 times per second.
21
A swimmer is treading water (with her head above the water) at the surface of a pool 3.00 m deep.
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 7 of 14
She sees a coin on the bottom directly below.
a) Is the apparent depth of the coin deeper than actual depth of the coin, the same as the
actual depth of the coin, or less than the actual depth of the coin?
b) How deep does the coin appear to be?
a) The apparent depth of the coin is less than the actual depth of the coin because of
refraction.
'
b) Because the swimmer is directly over the coin, the problem simplifies to TQUU . TQ,R V 'W Y =
X
(3.00) (1.00/1.33) = 2.26 m
22
The drawing shows two sheets of polarizing material. The transmission axis of one is vertical, and
that of the other makes an angle of 45° with the vertical. Unpolarized light shines on this
arrangement from the left and from the right. From which direction does at least some of the light
pass through both sheets?
A) From the left
B) From the right
C) From either direction
D) From neither direction
C.
23
Name the 3 primary colors of light.
red, green, and blue
24
The sketch below shows a wave pulse on a string moving from left to right towards the end which is
rigidly clamped. Sketch the wave pulse after the pulse encounters the boundary. Indicate direction
of wave pulse travel with an arrow in your sketch. What is the relationship of the speed, frequency,
wavelength, and amplitude of the incident wave pulse compared to the reflected pulse?
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 8 of 14
As the wave pulse approaches the fixed end, the
internal restoring forces which allow the wave to propagate exert an upward force on the end of the
string. But, since the end is clamped, it cannot move. According to Newton's third law, the wall must
be exerting an equal downward force on the end of the string. This new force creates a wave pulse
that propagates from right to left, with the same speed and amplitude as the incident wave, but with
opposite polarity (upside down).
25
The sketch below shows a wave pulse on a string moving from left to right towards the end which is
free to move vertically. Sketch the wave pulse after the pulse encounters the boundary. Indicate
direction of wave pulse travel with an arrow in your sketch. What is the relationship of the speed,
frequency, wavelength, and amplitude of the incident wave pulse compared to the reflected pulse?
The net vertical force at the free end must be
zero. The reflected wave pulse propagates from right to left, with the same speed and amplitude as
the incident wave, and with the same polarity (rightside up).
26
The figure shows a light ray striking a glass prism (n = 1.5). Explain/Draw the result of the ray striking
the edge of the prism as shown.
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
The glass‐air interface has a critical angle of 42°; sin Z, .
'J
'[
Page 9 of 14
; The angle of incidence in the 45‐45‐
prism is 45°, thus the light is totally reflected.
27
The sketch below is a snapshot of an incident wave on a string travelling from left to right from a low
density string to a higher density string. Sketch what happens after the pulse encounters the
boundary between the low and high density regions.
Part of the energy is reflected and part is
transmitted. The reflected wave pulse is inverted. The transmitted pulse has the same polarity as
the incident pulse. The amplitude of both pulses are smaller than the incident wave
28
The sketch below is a snapshot of an incident wave on a string travelling from left to right from a
higher density string to a lower density string. Sketch what happens after the pulse encounters the
boundary between the high and low density regions.
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 10 of 14
Part of the energy is reflected and part is
transmitted. The reflected wave pulse and the transmitted pulse have the same polarity as the
incident pulse. The amplitude of the transmitted pulse is larger than the incident pulse, but the
reflected pulse is smaller.
29
A person is viewing a ball through a Cyan filter. The ball appears to be green. The same
ball is viewed with a magenta filter and now appears to be black. What color is the ball
when revealed with white light?
Green
30
What colors of light combine to make
A) Yellow?
B) Magenta?
C) Cyan?
A) Green and Red
B) Red and Blue
C) Green and Blue
31 Two automobiles heading in opposite directions approach each other, the first at 88 ft/s and
the second at 66 ft/s. The driver of the first sounds a horn having a frequency of 400 Hz.
A) What frequency does the driver of the second car hear? ( Assume speed of sound to
be 1100 ft/s).
B) After the automobiles have passed each other what frequency does the drive of the
second car hear?
A) Use the Doppler‐effect equation:
2>
2@
= 1 A1
, where - 9 = velocity of listener, -* = velocity of source, - = speed of sound in
1?1
>
@
the medium. Since the listener is not moving, -9 = 0.
2>
=))
=))"##((D
=
/9 = #)#$
= 461 Hz
##))?(( ##))ABB
2
=))
>
B)##))A((
= ##))?BB
/9 =
=))"#)3=D
#)#$##BB
= 348 Hz
Crave the Wave ANSWER KEY
32
Mentor Invitational Feb 16, 2008
Page 11 of 14
Red light (0 . 664 \] ^\ -_`aa]D is used in Young’s experiment with the slits separated by a
distance T . 1.20 b 10A= m. The screen is located at a distance of L = 2.75 m from the slits. Find the
distance, y, on the screen between the central bright fringe and the third‐order bright fringe.
c
The Bright Fringes of a double slit; sin Z . ] O , m = 0,1,2,3,….
3"664 b 10AC ]D
Z . sinA#
. 0.951° e . 8 tan Z . f. fghi j
1.20 b 10A= ]
33
A cross‐sectional view of the eye is shown below:
Light entering the eye is shown in the diagram as rays. The picture demonstrates that a person with
myopia (near‐sightedness) will see a blurry image because the light is focused in front of the retina
and not on the retina.
A) What type of corrective lens will correct the problem?
B) Why?
C) Sketch the corrected light ray path.
A) Concave lens
B) The concave lens will cause the incoming light rays to diverge to compensate for the lens of
the eye.
C)
34
When a fish looks up at the surface of a perfectly smooth lake, the surface appears dark except
inside a circular area directly above it.
A) What optical phenomenon causes the rest of the surface to appear black?
B) Calculate the angle defining the illuminated region denoted by angle C in the picture below
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 12 of 14
a) When angle B is greater than the critical angle, there is total internal reflection.
b) By definition, Angle B is the critical angle when Angle is 90 degrees. Using Snell’s Law,
sin k .
35
'l mno C)
'p
.
"#.)D"#.)D
#.33
. 0.75
B = 48.6°°
C = (2)(B) = 97.2°
a) When light passes from a more optically dense medium into a less optically dense
medium, will it bend towards or away from the normal?
b) When light passes from a medium with a lower index of refraction into a medium with
a higher index of refraction, will it bend towards or away from the normal?
c) When light passes from a medium where the speed of light is slower into a medium
where the speed of light is faster, will it bend towards or away from the normal?
a) Away
b) Towards
c) Away
36
a) What are the regions of high pressure called in a sound wave?
b) What are the regions of low pressure called in a sound wave?
a) compressions
b) rarefactions
37
Two astronauts are 1.5 m apart in their spaceship. One speaks to the other. The conversation is
transmitted to earth via electromagnetic waves. The time it takes for sound waves to travel at 343
m/s through the air between the astronauts equals the time it takes for the electromagnetic waves
to travel to earth. How far away from the earth is the spaceship?
First find the time for the sound to travel between the astronauts.
O
#.K %
q . 1 . 3=3 %/* . 4.4 b 10A3 seconds
d = ct = (3.0 x 10B )( 4.4 b 10A3 D . r. s t rfi m
38
At what angle of incidence should a beam of light strike the surface of a still pond if the angle
between the reflected ray and the refracted ray be 90°? (n = 1.33 for water).
Angle of incidence = angle of reflection. Z& u Zv . 90°, wx Zv . 90° S Z& From Snell’s Law,
sin Z& . \ sin"90° ‐ Z& D . \ cos Z& , yw tan Z& . \, z{ . hs°
39
The sun is 149.6 million km from the earth.
a) How long does it take light from the sun to reach earth?
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 13 of 14
b) How long does it take sound to reach earth?
4
#=C.( | 109
a) Use x = vt
t = 1 = 3 b 108 = 498.666 seconds = 8.31 minutes
b) Sound can’t travel in the vacuum of space
40
One method of determining the refractive index of a transparent solid is to measure the critical angle
when the solid is in air. If Z, is found to be 38°, what is the index of refraction of the solid?
'
The critical angle, sin Z, . 'J where \# } \$ . Solve for \#
[
\$
1.00
\# .
.
. 1.62
sin Z, sin 38°
41
A fish in a still pond is observing the sunset. When the sun is setting and is at the horizon, where
does the fish perceive the sun to be? Give your answer in degrees from the horizontal. The index of
refraction for water is 1.33.
Use Snell’s law and the fact that this problem describes the water‐air interface critical angle.
"1.33D sin Z$ . 1, Z$ . sinA# 0.74 . 48.6°
The angle found is off the normal. The fish perceives the sun at 41.4° above the horizontal.
42
Sound waves with a frequency of 320 Hz are sent into the top of a vertical tube containing water at a
level that can be adjusted. If standing waves are produced at two successive water levels: 20 cm and
73 cm. What is the speed of the sound waves in the air of the tube?
The distance between the water levels is the distance between successive nodes, or half of a
wavelength.
0
. 73 S 20 . 53 `] _\T 0 . 106 `] . 1.06 ]
2
- . /0 = (320 )(1.06) = 339 m/s
43
A 40 cm long brass rod (cylindrical and hollow and open at both ends) is dropped one end first onto a
hard floor but is caught before it topples over. The resulting tone is measured by an oscilloscope and
is determined to have a frequency of 3 kHz. What is the speed of sound in brass?
c
Since the brass rod is open at both ends, the longitudinal wave will have L = n( ). Using the wave
1
1
$
equation, 2 . 0, substitute for 0. 8 . \"$2). For the fundamental frequency, n = 1.
L = 0.40 m
44
1
0.40 = "())))
~ . •gff j/€
A hunter is standing on the flat ground between two vertical cliffs that are directly opposite each
other. He is closer to one cliff than to the other. He fires a gun and , after a while, hears three
echoes. The second arrives 1.6 s after the first, and the 3rd arrives 1.1 s after the second. Assuming
the speed of sound is 343 m/s and that there are no reflections from the ground, find the distance
between the cliffs.
Let x be the distance from the hunter to the closer cliff and y the distance to the farther cliff.
$4
$L
$
q# = 1 , q$ = 1 , Δq . q$ S q# . 1.6 . 1 "e S bD. The 3rd echo arises from the second bouncing off
the closer cliff, thus q3 =
b.
"3=3D"#.#D
$
. 190 ]
$4
,
1
1RH
$
"3=3D"#.#?#.(D
solve for x, x =
e.
$
= 460 m.
Crave the Wave ANSWER KEY
Mentor Invitational Feb 16, 2008
Page 14 of 14
The total distance between the 2 cliffs is 650 m.
45
TRUE or FALSE:
a) The speed of sound in air is greater in warm air than in cool air.
b) The speed of sound along a guitar string is greater on a thicker string than on a
thinner string given that both strings are the same length and are under the same
tension.
In general, sound travels slowest in gases, faster in solids, and fastest in liquids
c)
a) True.
F
b) False. The speed of sound for a string is given by v . ‚ , where F is the tension on the
ƒ
string and u is the mass per unit length of the string, or linear density. The thinner string
has a smaller linear density and thus a greater speed.
c) False. In general, sound travels slowest in gases, faster in liquids, and fastest in
solids.