Question

A spaceship moves with a velocity of 5000 m/s. What is the relativistic factor \( \gamma \) for the spaceship? (Given that \( c = 3 \times 10^8 \, \text{m/s} \))

• A. \( 1.0001 \)
• B. \( 1.001 \)
• C. \( 1.0005 \)
• D. \( 1.00001 \)

Hint:

The relativistic factor \( \gamma \) becomes significantly greater than 1 only when the velocity is a substantial fraction of the speed of light. For velocities much smaller than \( c \), \( \gamma \) is nearly 1.

Answer 1

The Correct Option is A

The relativistic factor \( \gamma \) is calculated using the formula: \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \] Given: - Velocity \( v = 5000 \, \text{m/s} \) - Speed of light \( c = 3 \times 10^8 \, \text{m/s} \) Substituting these values yields: \[ \gamma = \frac{1}{\sqrt{1 - \frac{(5000)^2}{(3 \times 10^8)^2}}} \] This simplifies to: \[ \gamma = \frac{1}{\sqrt{1 - \frac{25 \times 10^6}{9 \times 10^{16}}}} = \frac{1}{\sqrt{1 - 2.78 \times 10^{-10}}} \] The approximate value of the relativistic factor is: \[ \gamma \approx 1.0001 \] Therefore, the relativistic factor for the spaceship is approximately \( 1.0001 \).