Two satellites, X and Y, move in circular orbits around a planet with orbital radii of r a… — JAMB UTME

Two satellites, $X$ and $Y$ , move in circular orbits around a planet with orbital radii of $r$ and $4r$ , respectively. If satellite $X$ travels at an orbital speed of $v$ , what is the orbital speed of satellite $Y$ ?

$\frac{v}{2}$ Answer
B
$\frac{v}{4}$
C
$2v$
D
$4v$

Answer

The orbital speed of satellite

Y

\frac{v}{2}

The orbital speed

v

of a satellite in a circular orbit of radius

r

around a planet of mass

M

is given by

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

. When the radius increases by a factor of 4 from

r

4r

, the new speed becomes

v_Y = \sqrt{\frac{GM}{4r}} = \frac{1}{2} \sqrt{\frac{GM}{r}} = \frac{v}{2}

Step-by-Step Solution

Write the formula for orbital speed of a satellite

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

Orbital speed depends on the gravitational mass

M

of the central body and the orbital radius

r

Set up the ratio between the orbital speeds of satellites

Y

and

X

\frac{v_Y}{v_X} = \sqrt{\frac{r_X}{r_Y}}

Since

G

and

M

are constant, orbital speed is inversely proportional to

\sqrt{r}

Substitute the given values

r_X = r

r_Y = 4r

, and

v_X = v

v_Y = v \sqrt{\frac{r}{4r}} = v \sqrt{\frac{1}{4}} = \frac{v}{2}

Evaluating the square root yields a factor of

\frac{1}{2}

Key Concept

Orbital Speed of a Satellite

Estimated Time:1m 0s

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