If due to air drag,the orbital radius of a sa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The orbital velocity of a satellite is given by $v = \sqrt{\frac{GM}{R}} = \sqrt{GM} \cdot R^{-1/2}$.Differentiating both sides with respect to $R

Vedclass · Accepted Answer

$\frac{\Delta R}{2} \sqrt{\frac{GM}{R^3}}$

Vedclass · Answer

(A) The orbital velocity of a satellite is given by $v = \sqrt{\frac{GM}{R}} = \sqrt{GM} \cdot R^{-1/2}$.Differentiating both sides with respect to $R$,we get:$\frac{dv}{dR} = \sqrt{GM} \cdot (-\frac{1}{2}) R^{-3/2} = -\frac{1}{2} \frac{\sqrt{GM}}{R \sqrt{R}} = -\frac{1}{2R} \sqrt{\frac{GM}{R}}$.For small changes,$\Delta v \approx \frac{dv}{dR} \Delta R$.Since the radius decreases by $\Delta R$,the change in radius is $-\Delta R$.Therefore,$\Delta v = (-\frac{1}{2R} \sqrt{\frac{GM}{R}}) \cdot (-\Delta R) = \frac{\Delta R}{2R} \sqrt{\frac{GM}{R}} = \frac{\Delta R}{2} \sqrt{\frac{GM}{R^3}}$.Thus,the change in orbital velocity is $\frac{\Delta R}{2} \sqrt{\frac{GM}{R^3}}$.

If due to air drag,the orbital radius of a satellite decreases from $R$ to $R - \Delta R$,where $\Delta R << R$,the expression for the change in orbital velocity $\Delta v$ is (mass of Earth is $M$):

Similar Questions

Two satellites are revolving around the Earth with orbital velocities $v_1$ and $v_2$ at radii $r_1$ and $r_2$ respectively,where $r_1 > r_2$. Then:

Two satellites $A$ and $B$ go round a planet $P$ in circular orbits having radii $4R$ and $R$ respectively. If the speed of the satellite $A$ is $3V$,the speed of the satellite $B$ will be ........ $V$.

Earth is moving around the sun in an elliptical orbit as shown. The ratio of $OB$ and $OA$ is $R$. Then the ratio of Earth's velocities at $A$ and $B$ is

$A$ satellite is moving around the earth with speed $v$ in a circular orbit of radius $r$. If the orbit radius is decreased by $1\%$,its speed will

Vedclass Test Series

Exam Paper Generator

Online Exam Module