Looking for classes? Ksquare Career Institute, Bengaluru →
A planet moving along an elliptical orbit is closest to the sun at a distance $r_1$ and farthest away at a distance of $r_2$. If $v_1$ and $v_2$ are the linear velocities at these points respectively, then the ratio $\frac{v_1}{v_2}$ is
A
$\frac{r_1}{r_2}$
B
$\left(\frac{r_1}{r_2}\right)^2$
C
$\frac{r_2}{r_1}$
D
$\left(\frac{r_2}{r_1}\right)^2$
Detailed Solution
The gravitational force of the sun on the planet is a central force, so its torque about the sun is zero and the angular momentum of the planet is conserved.
At the closest and farthest points the velocity is perpendicular to the radius vector, so L = mvr at both points.
$mv_1r_1 = mv_2r_2$
$v_1r_1 = v_2r_2$
$\frac{v_1}{v_2} = \frac{r_2}{r_1}$
At the closest and farthest points the velocity is perpendicular to the radius vector, so L = mvr at both points.
$mv_1r_1 = mv_2r_2$
$v_1r_1 = v_2r_2$
$\frac{v_1}{v_2} = \frac{r_2}{r_1}$