Question

Two spaceships are moving directly toward one another with a relative velocity of \(0.90 c .\) If an astronaut measures the length of his own spaceship to be \(30.0 \mathrm{m},\) how long is the spaceship as measured by an astronaut in the other ship?

Short answer

The length of the spaceship as observed by an astronaut in the other spaceship is approximately 13.08 meters.

Step-by-step solution

Identify the given information and the formula to use

We are given the relative velocity of the spaceships \(v = 0.90c\), the proper length of one spaceship \(L_0 = 30.0 m\), and we need to find the length as measured by an astronaut in the other ship. We will use the length contraction formula: \(L = L_0 \sqrt{1 - \frac{v^2}{c^2}}\), where \(L\) is the contracted length, \(L_0\) is the proper length, and \(v\) is the relative velocity.

Plug in the given values into the formula

Now, let's substitute the given values into the formula: \(L = 30.0 \sqrt{1 - \frac{(0.90c)^2}{c^2}}\)

Simplify and solve for L

Simplify the equation further: \(L = 30.0 \sqrt{1 - 0.81}\) Calculate the square root: \(L = 30.0 \sqrt{0.19}\) Finally, compute the length: \(L \approx 30.0 * 0.4359 \approx 13.08\)

Interpret the result

The length of the spaceship as measured by an astronaut in the other ship is approximately 13.08 meters. This shows that the length of the spaceship is contracted as observed from the other spaceship due to their relative velocity.