Question: Earth is moving around the sun in an elliptical orbit as shown. The ratio of OB and OA is R. then th...

Question

Earth is moving around the sun in an elliptical orbit as shown. The ratio of OB and OA is R. then the ratio of speed earth at A and B is:

A. $\mathop R\nolimits^{ - 1}$
B. $\sqrt R$
C. $R$
D. $\mathop R\nolimits^{2/3}$

Answer 1

Solution

HINT: In this we just try to understand the motion of earth around the sun and learn how gravitational effects this motion. Try to understand where we can use the conservation laws. Here we can apply the law of conservation of momentum. So, try to study about it.

Complete step-by-step answer:
The earth is moving around the sun in an elliptical orbit.
Given, the ratio of the distance from sun to earth in two positions A and B is
$\dfrac{{OB}}{{OA}} = R$
Now, we need to find the ratio of velocities of earth at the two points.
The earth and the sun are connected by gravitational force.
To find the ratio of velocities at the two points we can use the law of conservation of angular momentum, because in this system external torque is zero.
From, the law of conservation of angular momentum,
Angular momentum of earth at point A is equal to the angular momentum of earth at point B.
$m\mathop v\nolimits_a \mathop r\nolimits_a = m\mathop v\nolimits_b \mathop r\nolimits_b$
$\mathop v\nolimits_a \mathop r\nolimits_a = \mathop v\nolimits_b \mathop r\nolimits_b$
$\dfrac{{\mathop v\nolimits_a }}{{\mathop v\nolimits_b }} = \dfrac{{\mathop r\nolimits_b }}{{\mathop r\nolimits_a }}$
$\dfrac{{\mathop v\nolimits_a }}{{\mathop v\nolimits_b }} = \dfrac{{OB}}{{OA}}$
$\dfrac{{\mathop v\nolimits_a }}{{\mathop v\nolimits_b }} = R$
So, the ratio of velocity of earth at point A to the point B is R.
So, the correct option is (C)

NOTE: The earth orbits the sun in an elliptical orbit. The external torque on the earth in its orbit is zero. It is because gravitational force on the earth is towards the sun and the direction of position vector along the direction of earth to the sun. Since torque is defined as the cross product of the force and position vector and the force and position vector are parallel here the total external torque on earth is also zero.