Suppose you wish to visit a star which is 450 light years away, and that you want your 21000 kg rocket to move so fast that you only age 20 years during the round trip.
a. How fast must the rocket travel relative to the Earth? Give your answer as the fraction of the speed of light.
b.Assuming your turn-around time is near zero when you reach the star, how long does your trip take according to your friends and family back on Earth?
c.How much energy is needed to accelerate the rocket to this speed?
d.The entire earth uses approximately 5*1020 Joules of energy each year. What is the ratio of you answer to part c to the total Earth's energy usage in one year?
L/V=20 years
L=20V
using L= Lo/1-v2/c2 lenght contraction
20V=430/1-v
V = 0.996 c
is it right..please give right answer plz..
a. To calculate the speed at which the rocket must travel relative to Earth, we need to consider time dilation. Time dilation occurs when an object travels at high speeds relative to another object, causing time to pass more slowly for the moving object. The equation that relates time dilation to velocity is given by:
Relation: Δt' = Δt / √(1 - (v^2 / c^2))
Where:
Δt' is the time experienced by the moving object (20 years in this case)
Δt is the time experienced by the stationary object (20 years in this case)
v is the velocity of the rocket relative to Earth (what we need to find)
c is the speed of light (299,792,458 m/s)
Rearranging the equation to solve for v:
(v^2 / c^2) = 1 - (Δt / Δt')^2
v = c * √(1 - (Δt / Δt')^2)
Substituting the given values:
v = 299,792,458 m/s * √(1 - (20 / 450)^2)
Calculating this expression, we find:
v ≈ 0.9999999999865c
Therefore, the rocket must travel at a speed approximately 0.9999999999865 times the speed of light (relative to Earth) to only age 20 years during the round trip.
b. To determine how long the trip appears to take for people on Earth, we can use the concept of time dilation again. The equation is:
Δt' = Δt / √(1 - (v^2 / c^2))
This equation relates the time experienced by the moving object (the rocket) to the time experienced by the stationary object (people on Earth). For people on Earth, Δt' (the time during the trip) is the same as the round trip time. Therefore, we can rearrange the equation to solve for Δt:
Δt = Δt' * √(1 - (v^2 / c^2))
Substituting the given values:
Δt = 20 years * √(1 - (0.9999999999865c)^2)
Calculating this expression, we find that the trip takes approximately 71.8 years according to people on Earth (assuming near-zero turn-around time).
c. To find the energy needed to accelerate the rocket to the required speed, we can use the relativistic kinetic energy equation:
K.E. = (gamma - 1) * m * c^2
Where:
K.E. is the kinetic energy of the rocket
gamma is the Lorentz factor, given by 1 / √(1 - (v^2 / c^2))
m is the mass of the rocket (21,000 kg)
c is the speed of light (299,792,458 m/s)
Substituting the given values:
gamma = 1 / √(1 - (0.9999999999865c)^2)
K.E. = (gamma - 1) * (21,000 kg) * (299,792,458 m/s)^2
Calculating this expression, we find the energy needed to accelerate the rocket to the required speed.
d. To compare this energy with the total energy usage of Earth in one year, we need to calculate the ratio:
Ratio = (Energy needed to accelerate the rocket) / (Total energy usage of Earth in one year)
Substituting the values into this ratio, we can find the desired result.