A rocket launched from earth with a rest mass of 150,000Kg moves at 2.8x10^8 m/s during a voyage that takes 1.3 years, according to a clock on board the rocket.
a) Using relativity, calculate how long the voyage would take, from earth's reference frame.
b) Find the length of the rocket in earth's frame of reference when it is moving, given that the rest is 150m. Assume you are a stationary observer viewing the rocket from the side.
a) In order to calculate how long the voyage would take from Earth's reference frame, we need to use the concept of time dilation in special relativity. According to special relativity, time dilation occurs when an object is moving relative to an observer. The equation for time dilation is:
t = t_0 * γ
where t is the time measured by the observer on Earth, t_0 is the time measured by the clock on the rocket, and γ is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the rocket and c is the speed of light.
Given:
t_0 = 1.3 years = 1.3 * 365 days * 24 hours * 3600 seconds
v = 2.8 * 10^8 m/s
c = 3 * 10^8 m/s
First, let's calculate the Lorentz factor:
γ = 1 / sqrt(1 - (2.8x10^8)^2 / (3x10^8)^2)
= 1 / sqrt(1 - 0.31111)
= 1 / sqrt(0.68889)
= 1 / 0.829156
= 1.207877
Now, let's calculate the time measured by the observer on Earth:
t = t_0 * γ
= (1.3 * 365 * 24 * 3600) * 1.207877
≈ 3,427,554,182 seconds
Therefore, the voyage would take approximately 3,427,554,182 seconds (or around 108.6 years) from Earth's reference frame.
b) In order to find the length of the rocket in Earth's frame of reference, we need to use the concept of length contraction in special relativity. According to special relativity, length contraction occurs when an object is moving relative to an observer. The equation for length contraction is:
L = L_0 * √(1 - (v^2 / c^2))
where L is the length measured by the observer on Earth, L_0 is the rest length of the rocket, v is the velocity of the rocket, and c is the speed of light.
Given:
L_0 = 150 m
v = 2.8 * 10^8 m/s
c = 3 * 10^8 m/s
Let's calculate the length measured by the observer on Earth:
L = L_0 * √(1 - (2.8x10^8)^2 / (3x10^8)^2)
= 150 * √(1 - 0.31111)
≈ 150 * √(0.68889)
≈ 150 * 0.829156
≈ 124.3734 m
Therefore, the length of the rocket in Earth's frame of reference when it is moving, as observed by a stationary observer from the side, is approximately 124.37 meters.
a) To calculate how long the voyage would take from Earth's reference frame, we will use the concept of time dilation from special relativity. Time dilation states that the time experienced by an object moving relative to an observer is slower than the time experienced by the observer.
The equation we can use to calculate time dilation is:
t = t0 * √(1 - v^2/c^2)
Where:
t is the time experienced by the moving object (rocket)
t0 is the time experienced by the observer (on Earth)
v is the velocity of the rocket relative to Earth
c is the speed of light in a vacuum (3.00 x 10^8 m/s)
In this case, we are given:
t0 = 1.3 years = 1.3 * 365 * 24 * 60 * 60 seconds
v = 2.8 x 10^8 m/s
c = 3.00 x 10^8 m/s
Plugging these values into the equation, we can calculate t:
t = (1.3 * 365 * 24 * 60 * 60) * √(1 - (2.8 x 10^8 / 3.00 x 10^8)^2)
Simplifying this equation will give us the calculated time experienced by the moving object (rocket) from Earth's reference frame.
b) To calculate the length of the rocket in Earth's frame of reference, we will use the concept of length contraction from special relativity. Length contraction states that the length of an object moving relative to an observer appears shorter in the direction of motion.
The equation we can use to calculate length contraction is:
L = L0 * √(1 - v^2/c^2)
Where:
L is the length of the moving object (rocket) in Earth's frame of reference
L0 is the rest length of the rocket (given as 150 m)
v is the velocity of the rocket relative to Earth
c is the speed of light in a vacuum (3.00 x 10^8 m/s)
In this case, we are given:
L0 = 150 m
v = 2.8 x 10^8 m/s
c = 3.00 x 10^8 m/s
Plugging these values into the equation, we can calculate L:
L = 150 * √(1 - (2.8 x 10^8 / 3.00 x 10^8)^2)
Simplifying this equation will give us the length of the rocket in Earth's frame of reference.