Question

A mechanism on Earth used to shoot down geosyn- chronous satellites that house laser-based weapons is finally perfected and propels golf balls at 0.94c. (Geo- synchronous satellites are placed 3.58 x 10⁴ km above the surface of the Earth.) What is the distance from the Earth to the satellite, as measured by a detec- tor placed inside the golf ball?

Answer 1

Using the formula for length contraction in special relativity due to high velocity, the contracted length between the golf ball and the geosynchronous satellite is calculated to be approximately 12214.16 km.

Step-by-step explanation:

The question relates to the concept of length contraction in special relativity, which occurs because the golf ball is travelling at a significant fraction of the speed of light (0.94c). The length contraction formula, which falls under Einstein's theory of relativity, states:

L = L0 * sqrt(1 - v^2/c^2)

Where L0 is the rest length (3.58 x 104 km in this case), L is the contracted length, v is the velocity of the object, and c is the speed of light.

We can plug in the values to find the contracted length as follows:

L = 3.58 x 104 km * sqrt(1 - (0.94c)^2/c^2) = 3.58 x 104 km * sqrt(1 - 0.8836) = 3.58 x 104 km * sqrt(0.1164) ≈ 3.58 x 104 km * 0.3412 ≈ 12214.16 km

Therefore, the distance from the Earth to the satellite as measured by a detector placed inside the golf ball would be approximately 12214.16 km.