Question

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

• A. Increase by $1\%$
• B. Increase by $0.5\%$
• C. Decrease by $1\%$
• D. Decrease by $0.5\%$

Answer 1

The correct option is B

Increase by $0.5\%$

Step 1. Given data :

Speed $=v$

Radius $=r$

Orbit radius (decreased by) $=1\%$

Step 2. Find the speed of the satellite :

$v=\sqrt{\frac{GM}{r}}$

Where, $v$ is the velocity of a satellite in orbit of radius$r$, $G$ is the gravitational constant, and$M$ is the mass of the Earth,

Apply log on both sides,

$\log v=\frac{1}{2}\log G+\frac{1}{2}\log M-\frac{1}{2}\log r$

Differentiating on both sides,

$\frac{∆v}{v}=0+0-\frac{1}{2}\frac{∆r}{r}$

$\Rightarrow$$\frac{∆v}{v}=-\frac{1}{2}\frac{∆r}{r}$

Multiplying by $100$ on both sides,

$\frac{∆v}{v}\times 100=-\frac{1}{2}\frac{∆r}{r}\times 100$

$\mathrm{Change}\mathrm{in}\mathrm{orbital}\mathrm{velocity}\%=-\frac{1}{2}(\mathrm{Change}\mathrm{in}\mathrm{orbital}\mathrm{radius})\%$

$\mathrm{Change}\mathrm{in}\mathrm{orbital}\mathrm{velocity}\%=-\frac{1}{2}\times \left(-1\right)\%$

$\mathrm{Change}\mathrm{in}\mathrm{orbital}\mathrm{velocity}\%=0.5\%$ (positive value of percentage here shows the increase in the orbital velocity)

Hence, (B) is the correct option.