1. Classical principle of Relativity 2. Bohr’s Correspondence Principle
1.1 Starting with the defining equation for average velocity and assuming uniform translation acceleration, derive the equation Dx 5 v1D t 1
1⁄2a(D t)2. Solution:
For one-dimensional motion with constant acceleration, average ve-locity can be expressed as the arithmetic mean of the final veve-locity v2
and initial velocity v1. Assuming motion along the X-axis, we have v v
5 9 1
5 1 5
5 9
u R approaching E9 1
E approaching R
E stationary E receding from R 1
Ch. 1 Classical Transformations 28
and from the defining equation for average acceleration (Equation 1.9) we obtain
v25v11 aD t,
where the average sign has been dropped. Substitution of the second equation into the first equation gives
which is easily solved for Dx,
Dx 5v1D t 11⁄2a(D t)2.
1.2 Starting with the defining equation for average velocity and assum-ing uniform translation acceleration, derive an equation for the final ve-locity v2in terms of the initial velocity v1, the constant acceleration a, and the displacement Dx.
Answer: v225v211 2aDx
1.3 Do Problem 1.1 starting with Equation 1.5a and using calculus.
Solution:
Dropping the subscript notation in Equation 1.5a and solving it for dx gives
dx 5vdt.
By integrating both sides of this equation and interpreting v as the final velocity v2we have
Since v25v11 at, substitution into and integration of the last equa-tion yields
1.4 Do Problem 1.2 starting with Equation 1.7 and using calculus.
Answer: v225v211 2aDx
1.5 Staring with W 5 F ? Dx and assuming translational motion, show that W 5 DT by using the defining equations for average velocity and ac-celeration.
v v,
5 1 1
t
x a
2
1 1
D
D Dt
v . dx5 dt
x
x t t
2 0
1
2 =D
y y
v v .
5 1 5 1
x t t1 at dt t a t
0
1 2
D D 1 D
=D
^ h ^ h2
y
Problems 29
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Solution:
1.6 Starting with the defining equation for work (Equation 1.20) and using calculus, derive the work-energy theorem.
Answer: W 5 DT
1.7 Consider two cars, traveling due east and separating from one an-other. Let the first car be moving at 20 m/s and the second car at 30 m/s relative to the highway. If a passenger in the second car measures the speed of an eastbound bus to be 15 m/s, find the speed of the bus relative to ob-servers in the first car.
Solution:
Thinking of the first car as system S and the second as system S9, then
u 5 (30 2 20) m/s 5 10 m/s.
With the speed of the bus denoted accordingly as v9x5 15 m/s, vxis given by Equation 1.30a or Equation 1.31a as
vx5v9x1 u 5 15 m/s 1 10 m/s 5 25 m/s.
1.8 Consider a system S9 to be moving at a uniform rate of 30 m/s relative to system S, and a system S0 to be receding at a constant speed of 20 m/s relative to system S9. If observers in S0 measure the translational speed of a particle to be 50 m/s, what will observers in S9 and S measure for the speed of the particle? Assume all motion to be the positive x-direction
v
definition of a dot product assu g
average velocity definition
from Equation
Ch. 1 Classical Transformations 30
along the common axis of relative motion.
Answer: 70 m/s, 100 m/s
1.9 A passenger on a train traveling at 20 m/s passes a train station at-tendant. Ten seconds after the train passes, the attendant observes a plane 500 m away horizontally and 300 m high moving in the same direction as the train. Five seconds after the first observation, the attendant notes the plane to be 700 m away and 450 m high. What are the space-time coordi-nates of the plane to the passenger on the train?
Solution:
For the train station attendant
For the passenger on the train
1.10 From the results of Problem 1.9, find the velocity of the plane as measured by both the attendant and the passenger on the train.
Answer: 50 m/s at 36.9˚, 36.1 m/s at 56.3˚
1.11 A tuning fork of 660 Hz frequency is receding at 30 m/s from a sta-tionary (with respect to air) observer. Find the apparent frequency and wavelength of the sound waves as measured by the observer for vs5 330 m/s.
and
1.12 Consider Problem 1.11 for the case where the tuning fork is ap-proaching the stationary observer.
Answer: 726 Hz, 0.455 m
1.13 Consider Problem 1.11 for the case where the observer is approaching the stationary tuning fork.
Solution:
With n9 5 660 Hz and u 5 30 m/s for the case where R is approaching E9, we have
or l 5 l9 5 0.5 m.
1.14 Draw the appropriate schematic and derive the frequency transfor-mation equation for the case where the emitter E9 is stationary with respect to air and the receiver R is approaching the emitter.
Answer: n 5 n9 (1 1 k)
1.15 Consider Problem 1.11 for the case where the observer is receding from the stationary tuning fork.
Solution:
Given that n9 5 660 Hz, u 5 30 m/s, vs5 330 m/s, and R is receding from E9, then
or l 5 l9 5 0.5 m.
1.16 Consider a train to be traveling at a uniform rate of 25 m/s relative to stationary air and a plane to be in front of the train traveling at 40 m/s relative to and in the same direction as the train. If the engines of the plane produce sound waves of 800 Hz frequency, what is the frequency and wavelength of the sound wave to a ground observer located behind the
v Ch. 1 Classical Transformations
32
plane for vs5 335 m/s.
Answer: 670 Hz, 0.5 m
1.17 What is the apparent frequency and wavelength of the plane’s en-gines of Problem 1.16 to passengers on the train?
Solution:
Any stationary point in air between the plane and the train serves as a receiver of sound waves from the plane and an emitter of sound waves to the passengers on the train. Thus, from the previous problem we have n9 5 670 Hz, l9 5 0.5 m, u 5 25 m/s, and vs5 335 m/s, where the receiver (train) is approaching the emitter (stationary point). For this case the frequency becomes
and the wavelength is given by
1.18 A train traveling at 30 m/s due east, relative to stationary air, is ap-proaching an east bound car traveling at 15 m/s, relative to air. If the train emits sound of 600 Hz, find the frequency and wavelength of the sound to a passenger in the car for vs5 330 m/s.
Answer: 630 Hz, 0.5 m
1.19 A train traveling due west at 30 m/s emits 500 Hz sound waves while approaching a train station attendant. A driver of an automobile traveling due east at 15 m/s and emitting sound waves of 460 Hz is directly ap-proaching the attendant, who is at rest with respect to air. For vs5 330 m/s, find the frequency and wavelength of the train’s sound waves to the driver of the automobile.
Solution:
From the train to the attendant we have n9 5 500 Hz, u 5 30 m/s, vs
5 330 m/s, and E9 approaching R:
From the attendant to the automobile we have n9 5 550 Hz, u 5 15
670 720
5 9 11 5 Hz 11 5 Hz 67
n n ^ kh c 5m
v 0.5 0.5 .
5 1
5 5 m 5 9 5 m
u or
720 360 l s
n l l
v
/ ,
/ .
5 2 9 5
2 5
5 5 5
Hz Hz
Hz
m s m
1 1 1 11
500 550
550 330
5 3
s
n k
n
l n
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m/s, vs5 330 m/s, and R approaching E9:
1.20 After the automobile and train of Problem 1.19 pass the train station attendant, what is the frequency of the automobile’s sound waves to pas-sengers on the train?
Answer: 400 Hz
v
, .
5 9 1 5 1 5
5 9 5 5 1
5 5
Hz Hz
m or u m
1 550 1
22
1 575
5 3
575 375
5 3
s
n n k
l l l
n
^ h c m
Ch. 1 Classical Transformations 33a
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Introduction
The discussion of Galilean relativity in the last chapter was mostly in ac-cord with common sense. The results obtained were intuitive and in agree-ment with everyday experience for inertial systems separating from one another at relatively low speeds. The problems associated with comparing physical measurements made by different inertial observers were easily
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