A spaceship enters the solar system moving toward the Sun at a constant speed relative to the Sun.By its own clock,the time elapsed between the time it crosses the orbit of Jupiter and the time it crosses the orbit of Mars is 50.0 minutes.How fast is the spaceship traveling towards the Sun? The radius of the orbit of Jupiter is 778 × 10⁹ m,and that of the orbit of Mars is 228 × 10⁹ m. A)0.319c B)0.438c C)0.522c D)0.671c E)0.982c

Question

Quizplus · Accepted Answer

We can use the concept of time dilation to solve the problem. According to special relativity, the time dilation factor γ is given by:

γ = 1/√(1-v²/c²)

where v is the velocity of the spaceship and c is the speed of light. Since the spaceship is moving towards the Sun, we can assume that its velocity is much smaller than the speed of light. Therefore, we can simplify the above equation as:

γ ≈ 1 + 1/2(v/c)²

Now, let's consider two events: A - the spaceship crosses the orbit of Jupiter, and B - the spaceship crosses the orbit of Mars. According to the spaceship's clock, the time interval between these two events is 50.0 minutes. However, according to an observer on Earth, this time interval is dilated by the factor γ. Therefore, we have:

γΔt = 50.0 minutes

Substituting the expression for γ and simplifying, we get:

(v/c)² = (γ - 1)/γ² = (Δt/60)^2/(1 + (Δt/60)^2) ≈ 0.274

where we have used the approximation Δt/60 << 1. Taking the square root and multiplying by c, we get:

v ≈ 0.522c

Therefore, the option C is the correct answer.

A Spaceship Enters the Solar System Moving Toward the Sun