A rocket speeding by the Earth at 0.917c is roughly cylindrical in shape and oriented such that its axis is parallel to its velocity. At rest, it is measured to have a radius of 15 m and a length of 150 m. When speeding by, what is its volume as measured by observers on Earth? A) 6.78 × 10³ m³ B) 1.70 × 10⁴ m³ C) 4.23 × 10⁴ m³ D) 1.35 × 10⁴ m³ E) 1.06 × 10⁵ m³

Question

Quizplus · Accepted Answer

The volume of a cylinder can be calculated using the formula: V = πr²h, where r is the radius, h is the height and π is a constant.
In this case, the length of the cylinder is parallel to its velocity, so there will be no length contraction. However, the radius will undergo contraction in the direction of motion.
Using the formula for length contraction: 
r' = r/γ, where γ is the Lorentz factor given by γ = 1/√(1-v²/c²), where v is the velocity of the rocket and c is the speed of light.
Plugging in the numbers, we get:
γ = 1/√(1-0.917²) = 2.825
r' = 15/2.825 = 5.31 m
Now, we can calculate the volume as measured by observers on Earth using the following formula:
V' = π(r'²)h, where h is still the same as the length of the cylinder does not undergo contraction.
Plugging in the numbers, we get:
V' = π(5.31²)(150) ≈ 4.23 × 10¹⁴ m³
Therefore, the correct answer is (C).

A Rocket Speeding by the Earth at 0