In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit a star which is 410 ly away, and that you want your 24000 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.
0.999702691
b. As a check of your answer in part a, subtract your answer from 1 and input the result here.
0.000297309
c. 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? (you can use y for years)
820.243866 yr
d. How much energy is needed to accelerate the rocket to this speed?
e. The entire earth uses approximately 5*1020 Joules of energy each year. What is the ratio of you answer to part d to the total Earth's energy usage in one year?
I can't figure out part d and e
How do you get these answers?
To calculate the energy needed to accelerate the rocket to the required speed, we need to use Einstein's mass-energy equivalence formula: E = mc^2, where E is the energy, m is the mass, and c is the speed of light.
Given:
Mass of the rocket (m): 24000 kg
Speed (v): 0.999702691 times the speed of light (c)
a. How fast must the rocket travel relative to the Earth? Give your answer as the fraction of the speed of light.
Velocity (v) = 0.999702691c
b. As a check of your answer in part a, subtract your answer from 1 and input the result here.
Velocity (v) = 1 - 0.999702691c
c. 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? (you can use y for years)
To calculate the time experienced by friends and family back on Earth (t'), we can use the time dilation formula:
t' = t / sqrt(1 - v^2/c^2)
where t is the time experienced by the traveler, and v is the velocity relative to the speed of light.
t = 20 years (as given in the question)
v = 0.999702691c
t' = 20 / sqrt(1 - (0.999702691c)^2)
t' = 820.243866 years
d. How much energy is needed to accelerate the rocket to this speed?
To calculate the energy needed (E), we use the mass-energy equivalence formula:
E = mc^2
E = (24000 kg) * (299792458 m/s)^2
E = 2.03971461 x 10^24 Joules
e. The entire Earth uses approximately 5 * 10^20 Joules of energy each year. What is the ratio of your answer to part d to the total Earth's energy usage in one year?
Earth's energy usage per year: 5 * 10^20 Joules
Ratio = (Energy needed for the rocket) / (Earth's energy usage per year)
Ratio = (2.03971461 x 10^24 Joules) / (5 * 10^20 Joules)
Ratio = 4.079 * 10^3
So, the ratio is approximately 4.079 * 10^3.
To solve parts (d) and (e) of the problem, we need to consider the concept of kinetic energy and relativistic energy.
(d) The energy needed to accelerate the rocket to its required speed can be calculated using the relativistic kinetic energy equation:
\[KE = (\gamma - 1) \cdot mc^2\]
where:
- KE is the kinetic energy
- γ (gamma) is the Lorentz factor, given by \(γ = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)
- m is the mass of the rocket (24000 kg)
- c is the speed of light (approximately \(3 \times 10^8\) m/s)
To calculate the required energy, we need to find the velocity of the rocket (v) relative to Earth. We already know from part (a) that the fraction of the speed of light is 0.999702691, so we can put that into the equation to find v. Solving for v, we get:
\[v = 0.999702691c\]
Now we can substitute v into the equation for γ and calculate it:
\[γ = \frac{1}{\sqrt{1 - \frac{(0.999702691c)^2}{c^2}}}\]
Simplifying,
\[γ = \frac{1}{\sqrt{1 - 0.999407371}}\]
\[γ \approx 86.60254499\]
Finally, we can calculate the kinetic energy (KE):
\[KE = (86.60254499 - 1) \times (24000 \times (3 \times 10^8)^2)\]
\[KE \approx 6.708 \times 10^{21}\, \text{Joules}\]
Therefore, the energy needed to accelerate the rocket to this speed is approximately 6.708 × 10^21 Joules.
(e) To calculate the ratio of the answer to part (d) to the total Earth's energy usage in one year, we need to divide the answer to part (d) by the total Earth's yearly energy usage:
\[Ratio = \frac{6.708 \times 10^{21}}{5 \times 10^{20}}\]
Simplifying,
\[Ratio \approx 13.416\]
Therefore, the ratio of the energy required to accelerate the rocket to the total Earth's energy usage in one year is approximately 13.416.