Question

Assume you are agile enough to run across a horizontal surface at 14.70 m/s, independently of the value of the gravitational field. What would be (a) the radius of an airless spherical asteroid of uniform density 3.90x10³ kg/m³ on which you could launch yourself into orbit by running? (b) the mass of an airless spherical asteroid of uniform density 3.90x10³ kg/m³ on which you could launch yourself into orbit by running? (c) What would be the orbital period?

Answer 1

a)

by using the formula

G * M *m / R^2 = m * v1^2 / R

where M is the mass of the asteroid and R is the radius . since the mass of the asteroid may be writeen as

M = denisty * volume = rho * (4 * pie * R^2 / 3)

the requirement becomes

G * rho * ( 4 * pie * R^3 /3) * m /R^2 = m * (v1^2 / R)

R = sqrt( 3 * v1^2 / ( 4 *pie * G * rho)

the radius of the asteroid would then be

R = sqrt( (3 * 14.7^2) / ( 4 * 3.14 * 6.67 * 10^-11 * 3.9 * 10^3) = 14.08 km

b)

the mass of the asteroid will be

M = (rho * 4 * pie * R^3) / 3 = (3.9 * 10^3* 14086^3 * 4 * 3.14 ) / 3 = 4.56 * 10^16 kg

c)

T = 2 * pie / w

T = 2 * pie * R / v1

T = 2 * 3.14 * 14086 / 14.7 = 6017.7 sec