Question

In $2000$, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of $1\times {10}^{16}kg$and a radius of $8.8km.$

a. What is the speed of a satellite orbiting $5km$above the surface?

b. What is the escape speed from the asteroid?

Short answer

a. The speed of the satellite is $6.95m/s.$

b. The escape speed from the asteroid is$12.31m/s.$

Step-by-step solution

Given information a

Height = $5000m,$mass of asteroid (M) =$1.0\times {10}^{16}\mathrm{kg}$, Radius of asteroid (R) =$8800m.$

Calculation a

For a satellite to revolve around the planet the necessary centripetal force is provided by the gravitational force of attraction between the planet and the satellite.

$F_c=\text{centripetal force}=\frac{m{v}^2}{r}$

Where orbital radius of satellite r = R + h

And $F_g=\frac{GMm}{r^2}$

Now$$\begin{gathered} F_c=F_g \\ \frac{m{v}^2}{r}=\frac{GMm}{r^2} \\ v=\sqrt{\frac{GM}{r}}=\sqrt{\frac{GM}{R+h}} \end{gathered}$$

Substituting values and solving we get : $v=6.95\mathrm{m}/\mathrm{s}$.

Final answer a

The speed of the satellite is$6.95m/s.$

Given information b

Mass of the asteroid (M) = $1.0\times {10}^{16}\mathrm{kg},$radius of asteroid =$8800m.$

Calculation b

The escape speed from a planet is given by the formula : $v_e=\sqrt{\frac{2GM}{R}}$

Substituting values in the formula we get : $v_e=12.31\mathrm{m}/\mathrm{s}.$

Final answer b

The escape speed from the planet is$12.31m/s.$