Question

On Earth, froghoppers can jump upward with a takeoff speed of 2.8 m/s. Suppose you took some of the insects to an asteroid. If it is small enough, they can jump free of it and escape into space. Assume a typical asteroid density of 2.2 g/cm³. Suppose that one of the froghoppers jumped horizontally from a small hill on an asteroid.

a) What is the diameter (in kilometers) of the largest spherical asteroid from which they could jump free? (Express your answer to two significant figures.)

b) What would the diameter (in km) of the asteroid need to be so that the insect could go into a circular orbit just above the surface? (Express your answer to two significant figures.)

Answer 1

given
v = 2.8 m/s
rho = 2.2 g/cm^3

= 2.2*10^3 kg/m^3

let M is the mass and R is the radius of the asteroid.

density of the planet, rho = mass/volume

= M/(4/3*pi*R^3)

(4/3)*pi*rho = M/R^3

==> M/R^3 = (4/3)*pi*2.2*10^3

= 9215 kg/m^3

a) escape speed, ve = sqrt(2*G*M/R)

= sqrt(2*G*M*R^2/R^3)

ve = sqrt(2*G*M/R^3)*R

R = ve/sqrt(2*G*M/R^3)

= 2.8/sqrt(2*6.67*10^-11*9215)

= 2525 m

diameter, d = 2*R

= 2*2525

= 5050 m

= 5.05 km <<<<<<<<<<<---------------Answer

b) orbital speed, vo = sqrt(G*M/R)

= sqrt(G*M*R^2/R^3)

vo = sqrt(G*M/R^3)*R

R = ve/sqrt(G*M/R^3)

= 2.8/sqrt(6.67*10^-11*9215)

= 3571 m

diameter, d = 2*R

= 2*3571

= 7142 m

= 7.14 km <<<<<<<<<<<---------------Answer