To analyze the reasonableness of Chayse's results concerning the ultraviolet (UV) ray, we need to verify whether the calculated speed of light from his provided frequency and wavelength data aligns with the known speed of light in a vacuum.
Here are the steps we follow:
1. Identify the given values:
- Frequency ([tex]\( f \)[/tex]) of the UV ray: [tex]\( 1.53 \times 10^{16} \)[/tex] Hz
- Wavelength ([tex]\( \lambda \)[/tex]) of the UV ray: [tex]\( 1.96 \times 10^{-8} \)[/tex] meters
2. Calculate the speed of the UV ray:
The speed of a wave ([tex]\( c \)[/tex]) can be calculated using the formula:
[tex]\[
c = f \times \lambda
\][/tex]
Substituting the given values:
[tex]\[
c = (1.53 \times 10^{16} \, \text{Hz}) \times (1.96 \times 10^{-8} \, \text{m})
\][/tex]
Performing the multiplication:
[tex]\[
c = 1.53 \times 1.96 \times 10^{16} \times 10^{-8}
\][/tex]
[tex]\[
c = 2.9988 \times 10^8 \, \text{m/s}
\][/tex]
Thus, the calculated speed of the UV ray is approximately [tex]\( 299880000 \, \text{m/s} \)[/tex].
3. Compare with the known value:
- The speed of light in a vacuum is approximately [tex]\( 3.00 \times 10^8 \, \text{m/s} \)[/tex].
4. Evaluate the reasonableness:
We assess whether the calculated speed [tex]\( 299880000 \, \text{m/s} \)[/tex] is close enough to the known speed of light [tex]\( 3.00 \times 10^8 \, \text{m/s} \)[/tex].
One approach to this is to check the relative difference:
[tex]\[
\text{Relative difference} = \left| \frac{ \text{calculated speed} - \text{known speed} }{ \text{known speed} } \right|
\][/tex]
Substituting the values:
[tex]\[
\text{Relative difference} = \left| \frac{ 299880000 - 300000000 }{ 300000000 } \right|
\][/tex]
[tex]\[
\text{Relative difference} = \left| \frac{ -120000 }{ 300000000 } \right|
\][/tex]
[tex]\[
\text{Relative difference} = 0.0004
\][/tex]
This results in a relative difference of 0.04%, which is very small and within a reasonable tolerance level of 1%.
Hence, this minute discrepancy signifies that Chayse's findings are highly reasonable and accurately reflect the expected value of the speed of light in a vacuum for the given frequency and wavelength of the UV ray.
