(b) how far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1510 m/s? (c) with what speed will an object hit the asteroid if it is dropped from 981.8 km above the surface?

Answers:

(a)

(b) 397042.215 m

(c) 1917.76 m/s

Explanation:

The question is incomplete, please remember to write the whole question :) However, part (a) is written below:

(a) What is the escape speed on a spherical asteroid whose radius is  and whose gravitational acceleration at the surface is

Knowing this, let's begin:

a) In this part we need to find the escape speed on the asteroid:

(1)

Where:

is the universal gravitational constant

is the mass of the asteroid

is the radius of the asteroid

On the other hand we know the gravitational acceleration is , which is given by:

(2)

Isolating :

(3)

Substituting (3) in (1):

(4)

(5)

(6) This is the escape velocity

b) In this part we will use the Conservation of mechanical energy principle:

(7)

Being:

(8)

(9)

Where:

is the initial mechanical energy

is the final mechanical energy

is the initial kinetic energy

is the final kinetic energy

is the initial gravitational potential energy

is the final gravitational potential energy

is the mass of the object

is the radial speed of the object

is the distance above the surface of the object

Then:

(10)

Isolating :

(11)

(11)

(12) This is the distance above the asteroid's surface

c) We will use the Conservation of mechanical energy principle again, but now the condition is that the object is dropped at a distance . This means that at the begining the object only has gravitational potential energy and then it has kinetic energy and gravitational potential energy:

(13)

Isolating :

(14)

(15)

Finally: