Question

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 the red giant star Betelgeuse, which is 430 lyly away, and that you want your 20,000 kgkg rocket to move so fast that you age only 36 years during the round trip.

A. How fast (v) must the rocket travel relative to earth?
B. How much energy is needed to accelerate the rocket to this speed?
C. How many times larger is this energy than the total energy used by the United States in the year 2000, which was roughly 1.0 x 10^20 J?

Answer 1

a)
`v=0.999124c`

b)
`E=7.566*10^(22)`

c)
`E_a=760 times\ larger`

Step-by-step explanation:

From the question we are told that

Distance to Betelgeuse
`d_b=430ly`

Mass of Rocket
`M_r=20000`

Total Time in years traveled
`T_d=36years`

Total energy used by the United States in the year 2000
`E_(2000)=1.0*10^20`

Generally the equation of speed of rocket v mathematically given by

`v=(2d)/(\triangle t)`

`v=860ly/ \triangle t`

where

`\triangle t=(\triangle t')/((√(1-860/ \triangle t)^2))`

`\triangle t=(36)/((√(1-860/ \triangle t)^2))`

`\triangle t=√((860)^2+(36)^2)`

`\triangle t=860.7532`

Therefore

`v=(860ly)/( 860.7532)`

`v=0.999124c`

b)

Generally the equation of the energy E required to attain prior speed mathematically given by

`E=(1)/(√(1-(v/c)^2) )-1(20000kg)(3*10^8m/s)^2`

`E=7.566*10^(22)`

c)Generally the equation of the energy
`E_a` required to accelerate the rocket mathematically given by

`E_a=(E)/(E_(2000))`

`E_a=(7.566*10^(22))/(1.0*10^(20))`

`E_a=760 times\ larger`