A mass and the Earth are separated by a small distance. They are then released f

Question

A mass and the Earth are separated by a small distance. They are then released from rest. Assume that he acceleration is constant. When the mass has traveled a distance of x,

a. What is the ratio of the work done on the mass over the work done on the Earth?

b. What is the ratio of the magnitudes of the acceleration of the mass over the magnitude of the acceleration of the Earth?

c. What is the ratio of the magnitude of the velocity of the mass over the magnitude of the velocity of the Earth?

d. What is the ratio of the magnitude of the displacement of the mass over the magnitude of the displacement of the Earth?

Please provide your answer with details.

Explanation / Answer

a)

F = gravitational force of attaraction between mass and earth

d = distance travelled

since F is same on both mass and earth , hence the ratio of work done is given as

Wm /We = F x/Fx = 1

b)

r = initial distance between earth and the mass

ae = acceleration of earth = Gm/r2

am = acceleration of mass = GM/r2

ratio = am /ae = ( GM/r2 ) / (Gm/r2 ) = M/m

c)

Vfe = final velocity of earth after time "t" = ae t

Vfm = final velocity of mass after time "t" = am t

ratio = Vfm /Vfe = am t/(ae t) = M/m

d)

de = displacement of earth = (0.5) ae t2

dm = displacement of mass = (0.5) am t2

Ratio = dm /de = (0.5) am t2 / ((0.5) ae t2 ) = M/m

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