Question

Suppose the straight-line distance between New York and San Francisco is 4.1 × 106 m (neglecting the curvature of the earth). A UFO is flying between these two cities at a speed of 0.65c relative to the earth. What do the voyagers aboard the UFO measure for this distance?

Answer 1

Answer 2

Solution: Distance Measured by UFO Voyagers (Length Contraction in Special Relativity)

Given:

• Proper distance ($L_0$): $4.1\times 10^6\text{m}$ (Earth's frame).

• UFO speed ($v$): $0.65c$ relative to Earth.

• 
  

Step 1: Apply Length Contraction Formula

In special relativity, moving observers measure a contracted length:

$$L=L_0\sqrt{1-\frac{v^2}{c^2}}$$

Step 2: Plug in Values

$$L=(4.1\times 10^6\text{m})\sqrt{1-(0.65{)}^2}$$$$L=(4.1\times 10^6)\sqrt{1-0.4225}$$$$L=(4.1\times 10^6)\sqrt{0.5775}$$$$L\approx (4.1\times 10^6)\times 0.76\approx 3.116\times 10^6\text{m}$$

Final Answer:

The UFO voyagers measure the distance as:

$$\boxed{{\displaystyle 3.1\times 10^6\text{m}}}$$

1.