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Classical Doppler Shift Anyone who has watched auto racing on TV is aware of the Doppler shift. As a race car approaches the camera, the sound of its engine increases in pitch (frequency). After the car passes the camera, the pitch of its engine decreases. We could use this pitch to determine the relative speed of the car. This technique is used in many real world applications including ultrasound imaging, Doppler radar, and to determine the motion of stars.

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I. Moving Observer Assume that we have a stationary audio source that produces sound waves of frequency f and speed v. Source v Observer x y A stationary observer sees the time between each wave as

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I. Moving Observer If the observer is now moving at a velocity u relative to the source then the speed of the waves as seen by the observer is where the positive sign is when the observer is moving toward the source. The time between waves is now

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I. Moving Observer Taking the ratio of our two results we get that

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II. Moving Source We now consider the case in which the source is moving toward the observer. In this case, the wave's speed is unchanged but the distance between wave fronts (wavelength) is reduced. For the source moving away from the observer the wavelength increased. l Source u Observer x y uT v

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II. Moving Source

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Thus, the frequency seen by the observer for a moving source is given by

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Moving Observer Moving Source Note: The motion of the observer and source create different effects. For sound, this difference is explained due to motion relative to the preferred reference frame! This preferred frame is the reference frame stationary to the medium propagating the sound (air)!

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Relativistic Doppler Effect For light in vacuum the distinction between motion the source and detector can not be done. Therefore the expression we derived for sound can not be correct for light. Consider a source of light moving toward the detector with velocityv, relative to the detector. If source emits N electromagnetic waves in time Δt D (measured in the frame of the detector), the first wave will travel a distance cΔt D and the source will travel the distance v Δt D measures in the frame of detector. Consider a source of light moving toward the detector with velocity v, relative to the detector. If source emits N electromagnetic waves in time Δt D (measured in the frame of the detector), the first wave will travel a distance cΔt D and the source will travel the distance v Δt D measures in the frame of detector.

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Relativistic Doppler Effect The wavelength of the light will be: The frequency f ’ observed by the detector will therefore be:

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Relativistic Doppler Effect If the frequency of the source is f 0 it will emit N=f 0 Δt S waves in the time Δt S measured by the source. Then Here Δt S is the proper time interval ( the first wave and the N th wave are emitted at the same place in the source’s reference frame). Therefore times Δt S and Δt D are related as:

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Relativistic Doppler Effect When the source and detector are moving towards one another: - approaching - approaching - receding - receding

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Relativistic Doppler Shift Since light has no medium, there should be no difference between moving the source and the observer. The problem with our previous derivation when dealing with light is that we haven't considered that space and time coordinates are different for the source and observer. Thus, we must account for the contraction of space and dilation of time due to motion. After accounting for differences in time and space, we get the following result for both moving source and moving observer:

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A ball is thrown at 20.0 m/s inside a boxcar moving along the tracks at 40.0 m/s. What is the speed of the ball relative to the ground if the ball is thrown (a) forward (b) backward (c) out the side door?

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(a)(b)(c)

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A muon formed high in the Earth’s atmosphere travels at speed v = 0.990c for a distance of 4.60 km before it decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon live, as measured in its reference frame? (b) How far does the muon travel, as measured in its frame?

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A muon formed high in the Earth’s atmosphere travels at speed v = 0.990c for a distance of 4.60 km before it decays into an electron, a neutrino, and an antineutrino (a) How long does the muon live, as measured in its reference frame? (b) How far does the muon travel, as measured in its frame? For, (a) The muon’s lifetime as measured in the Earth’s rest frame is and the lifetime measured in the muon’s rest frame is.

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A muon formed high in the Earth’s atmosphere travels at speed v = 0.990c for a distance of 4.60 km before it decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon live, as measured in its reference frame? (b) How far does the muon travel, as measured in its frame? (b)

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The identical twins Speedo and Goslo join a migration from the Earth to Planet X. It is 20.0 ly away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same time on different spacecraft. Speedo’s craft travels steadily at 0.950c, and Goslo’s at 0.750c. Calculate the age difference between the twins after Goslo’s spacecraft lands on Planet X. Which twin is the older?

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In the Earth frame, Speedo’s trip lasts for a time: Speedo’s age advances only by the proper time interval during his trip.

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The identical twins Speedo and Goslo join a migration from the Earth to Planet X. It is 20.0 ly away in a reference frame in which both planets are at rest. The twins, of the same age, depart at the same time on different spacecraft. Speedo’s craft travels steadily at 0.950c, and Goslo’s at 0.750c. Calculate the age difference between the twins after Goslo’s spacecraft lands on Planet X. Which twin is the older? Similarly for Goslo, While Speedo has landed on Planet X and is waiting for his brother, he ages by: Then ends up older by.

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An atomic clock moves at 1 000 km/h for 1.00 h as measured by an identical clock on the Earth. How many nanoseconds slow will the moving clock be compared with the Earth clock, at the end of the 1.00-h interval?

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Solution:

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Solution:

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A spacecraft is launched from the surface of the Earth with a velocity of 0.600c at an angle of 50.0° above the horizontal positive x axis. Another spacecraft is moving past, with a velocity of 0.700c in the negative x direction. Determine the magnitude and direction of the velocity of the first spacecraft as measured by the pilot of the second spacecraft.

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Solution: Let frame S be the Earth frame of reference. Then

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A spacecraft is launched from the surface of the Earth with a velocity of 0.600c at an angle of 50.0° above the horizontal positive x axis. Another spacecraft is moving past, with a velocity of 0.700c in the negative x direction. Determine the magnitude and direction of the velocity of the first spacecraft as measured by the pilot of the second spacecraft. Solution: As measured from the S’ frame of the second spacecraft:

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A moving rod is observed to have a length of 2.00 m and to be oriented at an angle of 30.0° with respect to the direction of motion, as shown in Figure. The rod has a speed of 0.995c. (a) What is the proper length of the rod? (b) What is the orientation angle in the proper frame?

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is a proper length, related to by.

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A moving rod is observed to have a length of 2.00 m and to be oriented at an angle of 30.0° with respect to the direction of motion, as shown in Figure. The rod has a speed of 0.995c. (a) What is the proper length of the rod? (b) What is the orientation angle in the proper frame?.

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We have seen that Einstein’s postulates require important modifications in our ideas of simultaneity and our measurements of time and length. Einstein’s postulates also require modification of our concepts of mass, momentum, and energy We have seen that Einstein’s postulates require important modifications in our ideas of simultaneity and our measurements of time and length. Einstein’s postulates also require modification of our concepts of mass, momentum, and energy.

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Special Relativity & General Relativity

Special Relativity & General Relativity

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