Question

Many science fiction spaceships are powered by antimatter reactors. Suppose a $20\mathrm{m}$long spaceship, with a mass of $15000\mathrm{kg}$

when empty, carries $2000\mathrm{kg}$of fuel: $1000\mathrm{kg}$each of matter and antimatter. The matter and antimatter are slowly combined, and the energy of their total annihilation is used to propel the ship. After consuming all the fuel and reaching top speed, the spaceship flies past a space station that is stationary with respect to the planet from which the ship was launched. What is the length of the spaceship as measured by astronauts on the space station?

Short answer

The length is$17.69\mathrm{m}.$

Step-by-step solution

Given information

We have given,

length =$20\mathrm{m}$

We have to find the contracted length of the space ship.

Step 2; Simplify

we have to find the velocity of the rocket that we will find by relativistic energy,

$$\begin{gathered} \mathrm{E}={\mathrm{mc}}^2 \\ \mathrm{E}=2000\mathrm{kg}\times {(3\times {10}^8\mathrm{m}/\mathrm{s})}^2\mathrm{E}=18\times {10}^{19}\mathrm{J} \end{gathered}$$

Then energy is used to speed up the rocket is,

$$\begin{gathered} \mathrm{E}=\frac{{\mathrm{m}}_{\mathrm{ro}}{\mathrm{c}}^2}{\sqrt{1-{\displaystyle \frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}}} \\ \sqrt{1-\frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}=0.88 \end{gathered}$$

Then the length contraction formula is

$$\begin{gathered} \mathrm{L}\text{'}=\mathrm{L}\sqrt{1-\frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}} \\ \mathrm{L}\text{'}=(20\mathrm{m})(0.88) \\ \mathrm{L}\text{'}=17.69\mathrm{m} \end{gathered}$$