Question

You have been visiting a distant planet. Your measurements have determined that the planet’s mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large.

a. What is the planet’s radius?

b. To get back to earth, you need to escape the planet. What minimum speed does your rocket need?

Short answer

a. The radius of the planet is$1.8\times {10}^{7}m.$

b. The minimum speed that the rocket needs is$9.414km/s.$

Step-by-step solution

Part (a) Step 1 : Given information

The planet’s mass is twice that of the earth.

Part (a) Step 2 : Calculation 

Acceleration due to gravity is given by :

$$\begin{gathered} a=\frac{G\times M_p}{{R_p}^2}. \\ {R}_p=\sqrt{\frac{G\times M_p}{a}}. \end{gathered}$$

Where 'G' is the universal gravitational constant.

$M_P$is the mass of the planet.

$R_P$is the radius of the planet.

a is the acceleration due to gravity.

Part (a) Step 3 : Continuation of calculation 

Now calculating we get,

$$\begin{gathered} R_P=\sqrt{\frac{G\times M_p}{a}} \\ {R}_P=\sqrt{\frac{(6.67\times {10}^{-11}N{m}^2/k{g}^2)(2\times 5.98\times {10}^{24}kg)}{(9.81/4m/s^2)}} \\ {R}_P=\sqrt{3.252\times {10}^{14}{m}^2} \\ {R}_P=1.80\times {10}^7\mathrm{m} \end{gathered}$$

Part (a) Step 4 : Final answer 

The radius of the planet is$1.8\times {10}^{7}m.$

Part (b) Step 5 : Given information

Free-fall acceleration is one-fourth of the gravity of the earth.

Part (b) Step 6 : Calculation 

The formula for escape velocity is given by : $v=\sqrt{\frac{2G\times M_P}{R_P}}.$

Where 'v' is escape velocity.

Now,

localid="1648487351115" $$\begin{gathered} v=\sqrt{\frac{2(6.67\times {10}^{-11}N{m}^2/k{g}^2)(2\times 5.98\times {10}^{24}kg)}{(1.80\times {10}^{7}m)}} \\ v=9414\mathrm{m}/\mathrm{s} \\ \mathrm{v}=9.414\mathrm{km}/\mathrm{s}. \end{gathered}$$

Part (b) Step 7 : Final answer

The minimum speed that the rocket needs is$9.414km/s.$