Question

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

Short answer

The minimum speed that the probe needs to escape the solar system is$42.06km/s.$

Step-by-step solution

Given information 

The space station orbits the sun at the same distance as the earth i.e. the sun's radius.

Calculation

The expression for the minimum speed for the probe to escape from the solar system is given by according to the law of conservation of energy is : $v=\sqrt{\frac{2G\times M}{R}}.$

Where 'v' is the escape velocity.

'G' is the universal gravitational constant.

'M' is the mass of the sun.

'R' is the distance between sun and earth orbitlocalid="1648486344262" $=1.5\times {10}^{11}m.$

Continuation of calculation

Now, all the known values are substituted to find 'v' :

$$\begin{gathered} v=\sqrt{\frac{2G\times M}{R}} \\ v=\sqrt{\frac{2(6.67\times {10}^{-11}N{m}^2/k{g}^2)(1.989\times {10}^{30}kg)}{(1.5\times {10}^{11}m)}} \\ v=42058\mathrm{m}/\mathrm{s} \\ v=42.06\mathrm{km}/\mathrm{s}. \end{gathered}$$

Final answer

The probe needs to have a minimum speed of $42.06\mathrm{km}/\mathrm{s}$ to escape from the solar system.