Light takes 4 hours to travel the distance from sun to nepton. lf you travel in a spaceship at a speed 0.935 c(where c is the speed of light in vacumm), the time(in minutes) required to cover the same distance measured with a clock on the spaceship will be approximatelya)34b)56c)85d)120Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Time Dilation and the Speed of Light

Introduction:
In this question, we are given the distance between the Sun and Neptune and the time it takes for light to travel this distance. We are also given the speed of a spaceship relative to the speed of light. We need to determine the time it would take for the spaceship to cover the same distance as measured by a clock on the spaceship.

Time Dilation:
Time dilation is a phenomenon predicted by Einstein's theory of relativity. It states that time is not absolute and can dilate or contract depending on the relative motion between observers. When an object moves at speeds close to the speed of light, time appears to slow down for that object from the perspective of a stationary observer.

Calculating Time Dilation:
To calculate the time dilation experienced by the spaceship traveling at 0.935c, we can use the formula:

$t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$

Where:
t' = time measured on the spaceship
t = time measured by a stationary observer
v = velocity of the spaceship relative to the speed of light (0.935c)
c = speed of light in vacuum

Calculating the Time:
Given that the time taken for light to travel from the Sun to Neptune is 4 hours, we can substitute these values into the formula:

$t' = \frac{4}{\sqrt{1 - (0.935c)^2/c^2}}$

Simplifying the equation:

$t' = \frac{4}{\sqrt{1 - 0.935^2}}$

$t' = \frac{4}{\sqrt{1 - 0.874225}}$

$t' = \frac{4}{\sqrt{0.125775}}$

$t' = \frac{4}{0.354901}$

$t' \approx 11.28$ hours

Converting to Minutes:
To convert hours to minutes, we multiply the result by 60:

$t_{\text{minutes}} = 11.28 \times 60$

$t_{\text{minutes}} \approx 676.8$ minutes

Since the question asks for the approximate time, we can round the result to the nearest whole number:

$t_{\text{minutes}} \approx 677$ minutes

Conclusion:
The time required to cover the same distance as measured by a clock on the spaceship traveling at a speed of 0.935c is approximately 677 minutes. Therefore, the correct answer is option C.

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