Question

How fast must a rocket travel relative to the earth so that time in the rocket slows down to half its rate as measured by earth-based observers?.

Answer 1

In special relativity, time dilation occurs when an object is moving relative to an observer at a significant fraction of the speed of light. To determine how fast the rocket must travel relative to the earth for time in the rocket to slow down to half its rate as measured by earth-based observers, we can use the time dilation equation and solve for the velocity.

Step-by-step explanation:

In special relativity, time dilation occurs when an object is moving relative to an observer at a significant fraction of the speed of light. According to time dilation, the rate of time in the moving object slows down compared to the rate of time observed by earth-based observers. To determine how fast the rocket must travel relative to the earth for time in the rocket to slow down to half its rate as measured by earth-based observers, we can use the time dilation equation:

t' = t / sqrt(1 - v^2/c^2)

where t' is the time observed by earth-based observers, t is the time experienced in the rocket, v is the velocity of the rocket relative to the earth, and c is the speed of light.

Let's assume that t' is half of t. Plugging in these values into the equation, we get:

t/2 = t / sqrt(1 - v^2/c^2)

We can solve this equation for v:

v = c * sqrt(1 - 1/4) = c * sqrt(3/4) = c * sqrt(3) / 2

Therefore, the rocket must travel at a speed of c * sqrt(3) / 2 relative to the earth for time in the rocket to slow down to half its rate as measured by earth-based observers.

Answer 2

The rocket must travel at a speed of c√(3)/2 relative to the Earth so that time in the rocket slows down to half its rate as measured by Earth-based observers.

Step-by-step explanation:

In special relativity, time dilation occurs when an object is moving relative to an observer. According to the theory, time slows down for the moving object as its velocity approaches the speed of light. In this case, the question asks how fast a rocket must travel relative to the Earth so that time in the rocket slows down to half its rate as measured by Earth-based observers.

To determine the required velocity, we can use the time dilation equation: Δt' = Δt / √(1 - v^2/c^2), where Δt' is the time as measured by the Earth-based observers and Δt is the time as measured by the rocket. Since we want the time in the rocket to be half the time as measured by the Earth-based observers, we can set Δt' = 2Δt.

Plugging in the values, we get 2Δt = Δt / √(1 - v^2/c^2). Rearranging the equation, we have 2 = 1 / √(1 - v^2/c^2). Squaring both sides, we get 4 = 1 / (1 - v^2/c^2). Multiplying both sides by (1 - v^2/c^2), we get 4 - 4v^2/c^2 = 1. Simplifying the equation, we have 4v^2/c^2 = 3. Dividing both sides by 4, we get v^2/c^2 = 3/4. Taking the square root of both sides, we get v/c = √(3/4) = √(3)/2. Rearranging the equation, we get v = c√(3)/2.

Therefore, the rocket must travel at a speed of c√(3)/2 relative to the Earth so that time in the rocket slows down to half its rate as measured by Earth-based observers.