Let's break down this question into three parts, based on the sub-questions provided.
**a. What frequency does the baseball measure for the radiation from the radar gun?**
The frequency of the radar gun (f_0) is exactly 10 GHz (10^10 Hz or 10 billion Hertz) when the baseball is at rest. So, the baseball measures the frequency of the radar gun to be 10 GHz.
**b. What frequency does the radar gun measure for the radiation reflected from the baseball?**
The frequency that the radar gun measures for the radiation reflected from the baseball (f) depends on the speed of the baseball (v) due to the Doppler effect.
The Doppler effect states that an observer moving towards a source of waves will perceive the waves as having a higher frequency than they actually do. Since the baseball is moving towards the radar gun, the observed frequency (f) is higher than the emitted frequency (f_0).
This difference in frequencies is given by the formula:
f = f_0 * ((c + v) / c)
where `c` is the speed of light (3*10^8 m/s).
Rearranging terms, we find the baseball's speed (v) in terms of the observed and emitted frequencies:
v = c * (f/f_0 - 1)
This equation gives us the speed of the baseball provided we know the frequency for the radiation reflected from the baseball.
**c. If the radar gun can only resolve frequencies that differ by one part in 10⁸, what is the minimum resolution that it can provide (in mph) for the baseball's speed?**
Here, the minimum frequency difference the radar gun can resolve (delta_f) is given as f_0 / 10^8 which equals 100 Hz.
To find the minimum speed resolution of the radar gun, we set f = f_0 + delta_f and solve for the speed (v):
min_v = c * ((f_0 + delta_f)/f_0 - 1)
Substituting the given values, we find that the minimum speed the radar gun can resolve (min_v) is approximately 3 m/s.
Finally, to convert this speed into mph, we use the conversion factor 1 m/s = 2.23694 mph.
Therefore, the minimum speed resolution of the radar gun in terms of mph is approximately 6.71 mph.