To determine the rate at which the pulses are received on Earth from a spacecraft approaching at 0.6 times the speed of light (0.6c), we need to consider the effects of relativistic Doppler shift. This phenomenon occurs because the frequency of waves changes depending on the relative motion of the source and the observer.
Understanding the Doppler Effect
The relativistic Doppler effect can be described using the formula:
f' = f √((1 + β) / (1 - β))
where:
- f' is the observed frequency (rate of pulses received).
- f is the emitted frequency (rate of pulses transmitted).
- β is the ratio of the velocity of the source to the speed of light (v/c).
Calculating the Parameters
In this scenario:
- The emitted frequency, f, is 40 pulses/s.
- The speed of the spacecraft, v, is 0.6c, which gives us β = 0.6.
Applying the Formula
Now, we can substitute these values into the Doppler effect formula:
f' = 40 √((1 + 0.6) / (1 - 0.6))
Calculating the terms inside the square root:
- 1 + 0.6 = 1.6
- 1 - 0.6 = 0.4
This gives us:
f' = 40 √(1.6 / 0.4)
Now, simplifying the fraction:
1.6 / 0.4 = 4
Thus, we have:
f' = 40 √4
Since the square root of 4 is 2, we find:
f' = 40 * 2 = 80 pulses/s
Final Result
The rate at which the pulses are received on Earth is therefore 80 pulses/s. This corresponds to option C in the choices provided.
