Question

If our moon was twice as far away and had twice as much mass it has now, how would its new speed compare to its present speed?
A. The new speed would be half as much as the present speed
B. The new speed would be smaller, but more than half the present speed
C. The new speed would be one-fourth the present speed
D. The new speed would be greater than the present speed
E. The new speed would be the same as the present speed

Answer 1

Hint: The orbital velocity of an astronomical object around a body is independent of its own mass. It depends on the mass of the object or planet that it is orbiting and its distance from the said planet/object.

Formula used:
$v = \sqrt {\dfrac{{GM}}{r}} $, where v is the orbital speed, G is the universal gravitational constant, M is the mass of the planet (in our case the Earth), and r is the distance between the object and the Earth.

Complete step by step answer
The orbital speed of an astronomical body or object is the speed at which it orbits around the more massive object relative to the centre of mass.
In this question, we are required to find the new orbital speed for the moon when its physical parameters relative to the Earth are changed a bit. The data given to us includes:
Initial mass of the moon: m
New mass of the moon: 2m
Initial distance of the moon: r
New distance of the moon: 2r
The initial and the new mass of the Earth remain constant, according to the question.
We know that the initial orbital speed will be given as:
$v = \sqrt {\dfrac{{GM}}{r}} $
Now, we substitute the new values to get the new orbital speed as:
${v_{new}} = \sqrt {\dfrac{{GM}}{{2r}}} = \dfrac{v}{{\sqrt 2 }} = 0.707v$ [As M is the mass of the Earth]
Hence, the new speed is lesser than the original speed of the moon but more than half the original speed.

Thus, the correct answer is option B.

Note The actual orbital speed of the moon around the Earth is $1.022km/s$. It is the speed required to achieve the balance between the inertia of the moon's motion and the gravity's pull on the moon.