What would be the gamma factor for a spacecraft traveling at 7,800 m/s?

 0.85 1.0 1.1 1.8

The gamma factor (\(\gamma\)) in the context of special relativity is given by the Lorentz factor, which is calculated using the equation:

\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

where:
- \( v \) is the velocity of the object (in meters per second, m/s)
- \( c \) is the speed of light in a vacuum (\( c \approx 3 \times 10^8 \, \text{m/s} \))

Given:
- \( v = 7,800 \, \text{m/s} \)

First, calculate the fraction \(\frac{v^2}{c^2}\):

\[ \frac{v^2}{c^2} = \left(\frac{7,800}{3 \times 10^8}\right)^2 \]
\[ \frac{v^2}{c^2} = \left(\frac{7,800}{300,000,000}\right)^2 \]
\[ \frac{v^2}{c^2} = \left(2.6 \times 10^{-5}\right)^2 \]
\[ \frac{v^2}{c^2} = 6.76 \times 10^{-10} \]

Next, substitute this value into the Lorentz factor formula:

\[ \gamma = \frac{1}{\sqrt{1 - 6.76 \times 10^{-10}}} \]

Since \( 6.76 \times 10^{-10} \) is a very small number, we can approximate:

\[ \gamma \approx \frac{1}{\sqrt{1 - 0}} \]
\[ \gamma \approx 1 \]

Thus, the gamma factor for a spacecraft traveling at 7,800 m/s is very close to 1.

Therefore, the correct answer is:
\[ 1.0 \]

Would you like further details or have any questions?

Here are 8 related questions:

1. What is the Lorentz factor and how is it used in special relativity?
2. How does velocity affect the gamma factor in special relativity?
3. Why does the gamma factor approach 1 as velocities become much less than the speed of light?
4. What are the implications of a gamma factor greater than 1 in relativistic physics?
5. How does time dilation relate to the gamma factor?
6. What is the significance of the speed of light in the Lorentz factor equation?
7. How would the gamma factor change if the spacecraft were traveling at 90% of the speed of light?
8. Can you explain the concept of relativistic mass increase with the gamma factor?

**Tip:** In relativistic calculations, for velocities much less than the speed of light, the gamma factor remains close to 1, simplifying many calculations.

Learn how to calculate the gamma factor (Lorentz factor) for a spacecraft traveling at 7,800 m/s using the Lorentz factor formula in special relativity. Understand the implications of relativistic speeds and the approximation of gamma factor near 1. Suitable for advanced physics enthusiasts.

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