Construct Your Own Problem

On February 14, 2000, the NEAR spacecraft was successfully inserted into orbit around Eros, becoming the first artificial satellite of an asteroid. Construct a problem in which you determine the orbital speed for a satellite near Eros. You will need to find the mass of the asteroid and consider such things as a safe distance for the orbit. Although Eros is not spherical, calculate the acceleration due to gravity on its surface at a point an average distance from its center of mass. Your instructor may also wish to have you calculate the escape velocity from this point on Eros.

Solution 50PE

Step 1 of 5</p>

A spacecraft is inserted into orbit of asteroid Eros of mass kg . If the radius of the orbit of the satellite is 50 km then find a.) Calculate the orbital speed. b.) Although Eros is not spherical ,calculate acceleration due to gravity on its surface at a point an average distance from its centre of mass. c.) Calculate the escape velocity.

a.)

Step 2 of 5</p>

The orbital speed of the spacecraft can be calculated using the expression which relates the mass of the spacecraft and radius of the orbit.

Where, = orbital velocity in m/s

G = Gravitational constant = 6.67410-11m3 kg-1 s-2

R = radius of the orbit in m

M = mass of the asteroid in kg

Given : R = 50 km = 50,000 m, M = kg

=

=

=

= 2.945 m/s

Therefore, the orbital speed of the spacecraft is 2.945 m/s

b.)

Step 3 of 5</p>

The acceleration due to gravity on its surface can calculated using the expression which relates mass of the asteroid and radius of the asteroid.

where, g = acceleration due to gravity in m/s2, M = mass of the asteroid in kg, r = radius of the asteroid in km

Now, the radius of the asteroid can be estimated as follows

Mass (ME) of the Earth is proportional to cube of its radius(RE)

= 5.972 1024 kg, RE= 6371 km

So, where, RA and MA are radius and mass of the asteroid respectively.

Therefore, =

=

=

= (6371)

= 6.5 km