asteroid and planet

Question

Consider the motion of an asteroid with mass m under the gravitational force applied by a planet with mass M. We assume that the planet is at rest and M>m. At an initial instant (say, at time=0 ), the asteroid is at a distance of 4R to the center of the planet and has initial speed given by v_0 where v_0=sqrt(GM/R)

a.) Find the total mechanical energy of the asteroid.

Does the asteroid escape from the planet or does it do a bounded motion?

(c) Which geometric shape best describes the trajectory followed by the asteroid?

Can you please help me with this question?

Answer 1

Problem:

Asteroid and Planet

Consider the motion of an asteroid with mass m under the gravitational force applied by a planet with mass M. We assume that the planet is at rest and M>m. At an initial instant (say, at time=0 ), the asteroid is at a distance of 4R to the center of the planet and has initial speed given by v_0 where v_0=sqrt(GM/R)

a.) Find the total mechanical energy of the asteroid.

b.) Does the asteroid escape from the planet or does it do a bounded motion?

c.) Which geometric shape best describes the trajectory followed by the asteroid?

Solution:

Given:

m = mass of asteroid

M = mass of planet

t = 0

R = radius of planet

r = radius of orbit = 4R

v0 = v = SQRT(GM/R)

G = 6.67 x 10^-11

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a.) Mechanical energy, E

E = K + U

K = (1/2)mv^2

K = (1/2)m(SQRT(GM/R))^2

K = GMm/2R total kinetic energy

U = -GMm/r

U = -GMm/4R total potential energy

E = K + U

= GMm/2R + GMm/4R

= [2GMm + GMm]/4R

= 3GMm/4R

= (3/4)*GMm/R

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b.) Escapes?

Let GMm/R = k a constant

K = GMm/2R = k/2

U = GMm/4R = k/4

Since the kinetic energy is twice as large as the potential energy, the asteroid escapes from the planet

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c.) Orbit shape

Since the asteroid escapes, the geometric shape of the asteroid is a parabola.

If it did orbit the planet (it does not), the geometric shape would most likely be elliptical.