On Earth, froghoppers can jump upward with a takeoff speed of 2.8 m/s. Suppose y

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. (a) What is the diameter (in km) of the largest spherical asteroid from which they could jump free? Assume typical asteroid denisty of 2.0 g/cm^3. (b) Suppose that one of the froghoppers jumped horizontally from a small hill on an asteroid. What would the diameter (in km) of the ateroid need to be so that the insect could go into a circular orbit just above the surace?

Explanation / Answer

a)

escape means r-> infinity and v= 0

1/2 mv^2 - G M m/r = 0

M = d 4/3 pi r^3

0.5*2.8^2 - 6.67E-11*(2000*4/3*pi*r^3)/r = 0

r=2648 m

d = 5296 m

b)

to go in an orbit

G M m/r^2 = m v^2/r

G M/r = v^2

6.67E-11*(2000*4/3*pi*r^3)/r = 2.8^@

r=3746 m

so d = 7492 m

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