A spaceship moving at constant speed u relative to us broadcasts a radio signal at constant frequency f0. As the spaceship approaches us, we receive a higher frequency f; after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not f0, and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than f0? (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus f equals 1/T. Use the time dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency f0 = 345 MHz as measured in a frame moving with the ship. The spaceship is moving at a constant speed 0.758c relative to us. What frequency f do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, f - f0? (c) Use the result of part (a) to calculate the frequency f and the frequency shift (f - f0) we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?
A spaceship moving at constant speed u relative to us broadcasts a radio signal at constant frequency f0. As the spaceship approaches us, we receive a higher frequency f; after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not f0, and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than f0? (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus f equals 1/T. Use the time dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency f0 = 345 MHz as measured in a frame moving with the ship. The spaceship is moving at a constant speed 0.758c relative to us. What frequency f do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, f - f0? (c) Use the result of part (a) to calculate the frequency f and the frequency shift (f - f0) we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?
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