Question

You stand on a spherical asteroid of uniform density whose mass is $\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{16}}\mathit{K}\mathit{g}$and whose radius is 10Km. These are typical values for small asteroids, although some asteroids have been found to have much lower average density and are thought to be loose agglomerations of shattered rocks.

(a) How fast do you have to throw the rock so that it never comes back to the asteroid and ends up traveling at a speed of 3 m/swhen it is very far away?

(b) Sketch graphs of the kinetic energy of the rock, the gravitational potential energy of the rock plus asteroid, and their sum, as a function of separation (distance from centre of asteroid to rock).

Label the graphs clearly. The asteroid, and their sum, as a function of separation (distance from centre of asteroid to rock).

Label the graphs clearly.

Short answer

(a) The rock's initial speed must be 16.6 m/s .

(b) The graph of kinetic energy of rock, the graph of kinetic energy of rock,

The graph of the gravitational potential energy of a rock,

The graph of the sum of gravitational potential energy of rock, asteroid and the kinetic energy of the rock is,

Step-by-step solution

Given Data

Mass is$2\times {10}^{16}kg$.

Radius is 10Km.

Concept of the escape speed

The following equation gives the object's escape speed from the planet:

Where Gis the gravitational constant, Mis the mass of the planet and Ris the radius of the planet.

Determine the initial speed

(a)

(1) The formula for the escape speed of an object from a certain planet can be expressed as,

$V_e=\sqrt{\frac{2GM}{R}}$

Where, G is the gravitational constant with a value of$6.67\times {10}^{-11}N.m^2/k{g}^2$

M is the planet's mass

R is the planet's radius

(2) To get the final kinetic energy, add the value of the gravitational potential energy to the initial kinetic energy of the object. It can be shown as,

${\mathrm{K}}_i+{\mathrm{U}}_i={\mathrm{K}}_f$

Derive the formula above to get the equation for the initial speed.

Plot the graphs of kinetic energy, gravitational potential energy and their sum

(b)

From the preceding section, it can be concluded that as the distance between the asteroid and the rock rises, the rock's speed decreases. When a result, as the distance between the asteroid and the rock rises, the kinetic energy of the rock decreases. The graph is given.

The following is a graph of the gravitational potential energy of a rock and an asteroid:

The following graph depicts the sum of gravitational potential energy of rock and asteroid, as well as the kinetic energy of the rock: