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/cm^{3}. 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**

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