Math solution on Space contraction

Space contraction, also known as length contraction, is another concept from Einstein's theory of relativity. According to this theory, an object moving at a high velocity will appear shorter in the direction of motion to an observer at rest compared to its length when measured in its own rest frame.

The formula for length contraction is given by:

$${𝐿}^{\mathrm{\prime }}=𝐿\sqrt{1-\frac{{𝑣}^2}{{𝑐}^2}}$$

where:

• ${𝐿}^{\mathrm{\prime }}$ is the contracted length (the length observed by an observer at rest relative to the moving object).
• $𝐿$ is the proper length (the length measured by an observer moving with the object, i.e., at rest relative to the object).
• $𝑣$ is the relative velocity between the observer and the moving object.
• $𝑐$ is the speed of light in a vacuum ($𝑐\approx 3\times 10^8\text{m/s}$).

Example Problem

Let's solve an example problem step-by-step using this formula.

Problem

A spaceship travels at a speed of $2.4\times 10^8\text{m/s}$ relative to an observer on Earth. The length of the spaceship as measured by an observer on the spaceship (proper length) is 100 meters. What is the length of the spaceship as measured by the observer on Earth?

Solution

1. Identify the given values:

   • Relative velocity, $𝑣=2.4\times 10^8\text{m/s}$
   • Proper length, $𝐿=100\text{m}$
   • Speed of light, $𝑐=3\times 10^8\text{m/s}$

2. Substitute the values into the length contraction formula:

   $${𝐿}^{\mathrm{\prime }}=𝐿\sqrt{1-\frac{{𝑣}^2}{{𝑐}^2}}$$
   $${𝐿}^{\mathrm{\prime }}=100\text{m}\sqrt{1-\frac{(2.4\times 10^8\text{m/s}{)}^2}{(3\times 10^8\text{m/s}{)}^2}}$$

3. Calculate the term inside the square root:

   $$\frac{{𝑣}^2}{{𝑐}^2}=\frac{(2.4\times 10^8\text{m/s}{)}^2}{(3\times 10^8\text{m/s}{)}^2}=\frac{5.76\times 10^{16}}{9\times 10^{16}}=\frac{5.76}{9}=0.64$$

4. Continue with the calculation:

   $${𝐿}^{\mathrm{\prime }}=100\text{m}\sqrt{1-0.64}=100\text{m}\sqrt{0.36}=100\text{m}\times 0.6$$
   $${𝐿}^{\mathrm{\prime }}=60\text{m}$$

So, the length of the spaceship as measured by the observer on Earth is 60 meters.

Conclusion

Using the length contraction formula, we found that for an observer on Earth, a spaceship moving at $2.4\times 10^8\text{m/s}$ will appear to be 60 meters long, even though its proper length (the length measured by an observer on the spaceship) is 100 meters. This illustrates how high velocities can significantly affect the perceived length of an object.