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The problem asks to determine the limit used to find the force applied by the motor as the spaceship approaches a velocity close to \( 3 \times 10^8 \, \text{m/s} \).

The force is given by the equation:

\[
F(v) = \frac{98100}{\sqrt{1 - \frac{v^2}{9 \times 10^{16}}}}
\]

### Analysis:

As \( v \) approaches \( 3 \times 10^8 \, \text{m/s} \), the term \( \frac{v^2}{9 \times 10^{16}} \) approaches 1. This makes the denominator of the equation \( \sqrt{1 - \frac{v^2}{9 \times 10^{16}}} \) approach 0, which causes the force \( F(v) \) to approach infinity.

This behavior is consistent with relativistic mechanics, where an infinite amount of force would be required to accelerate an object to the speed of light.

### Limit Calculation:

The question specifically asks about the limit of \( F(v) \) as \( v \) approaches \( 3 \times 10^8 \, \text{m/s} \):

\[
\lim_{v \to 3 \times 10^8} F(v) = \lim_{v \to 3 \times 10^8} \frac{98100}{\sqrt{1 - \frac{v^2}{9 \times 10^{16}}}}
\]

As \( v \) approaches \( 3 \times 10^8 \, \text{m/s} \), the denominator \( \sqrt{1 - \frac{v^2}{9 \times 10^{16}}} \) approaches zero, causing \( F(v) \) to approach infinity.

### Answer:

The correct answer is:

**C.** \(\lim_{v \to 3 \times 10^8} F(v)\)

This answer correctly represents the force tending towards infinity as the velocity approaches the speed of light.

If you have more questions or need further details, feel free to ask!

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**Here are five relative questions to deepen your understanding:**

1. What does the relativistic factor in the equation represent in physical terms?
2. How does relativistic mass affect the force required to accelerate an object near the speed of light?
3. What is the significance of the speed of light as a limit in relativistic physics?
4. How does the force function \( F(v) \) change for lower velocities, far from the speed of light?
5. What would be the physical implications if the force \( F(v) \) did not tend to infinity?

**Tip:** When dealing with relativistic equations, always consider how terms behave as velocity approaches the speed of light.

Explore the calculation of force in relativistic mechanics as velocity approaches 3 × 10^8 m/s. Learn about the equation used, its implications in physics, and why force tends to infinity near the speed of light. Suitable for advanced physics enthusiasts.

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