Question

derivation- orbital velocity

PLZ HELP ME TO DO DERIVATION OF ORBITAL VELOCITY.

NOTE:-

(PLZ DON'T SPAM OR COPY FROM GOOGLE DO IT YOUR OWN.

and plz exam whole orbital velocity
and plz do on notebook then send pic of the answer. do briefly not in short.

IF YOU DONT KNOW PLZZ DONT PRANK.
LET'S GIVE CHANCE TO OTHERS FOR THE QUESTIONS.)

(TODAY IS MY PHYSICS EXAM )
​

Answer 1

Answer:

To Derive :

Formula for orbital Velocity.

Concept:

Orbital Velocity is a Velocity used to define the trajectory and Velocity of natural and artificial satellites going around celestial bodies.

Examples:

• Telecommunication Satellites
• Navigation satellites

Proof:

When a satellite is revolving around another celestial body , the Centripetal force is attributed to the gravitational force between the 2 bodies.

Let mass of satellite be m , it's Velocity be v , mass of the planet be M , Universal Gravitational constant be G and radius of the planet be r

$\therefore \: centripetal \: f = gravitation \: f$

$= > \dfrac{m {v}^{2} }{r} = \dfrac{GmM}{ {r}^{2} }$

Cancelling the equal terms on both sides , we get :

$= > {v}^{2} = \dfrac{GM}{ r }$

$= > v = \sqrt{\dfrac{GM}{ r }}$

This is the orbital Velocity of the satellite.

Points to note:

• Orbital Velocity at a certain height doesn't depend on the mass of the satellite
• It depends on the mass of planet.

Answer 2

$\underline{\boxed{ \bold{ \huge{ \purple{Answer}}}}} \\ \\ \\ \star \sf \: \underline{\boxed{ \red{\bold{To \: Derive \: :}}}} \\ \\ \implies \sf \: orbital \: velocity \: of \: satellite \\ \\ \star \sf \: \underline{\boxed{ \bold{ \red{Definition \: :}}}} \\ \\ \implies \sf \: The \: velocity \: with \: which \: a \: satellite \\ \sf \: moves \: in \: its \: closed \: orbit \: is \: called \: orbital \\ \sf \: velocity \: of \: satellite\\ \\ \star \sf \: \underline {\boxed{ \bold{ \red{Derivation \: :}}}} \\ \\ \implies \sf \: let \: the \: satellite \: of \: mass \: m \: is \: revolving \\ \sf \: around \: a \: planet \: of \: mass \: M \: with \: velocity \: V \\ \\ \implies \sf \: radius \: of \: planet \: is \: R \: and \: gravitational \\ \sf \: constant \: is \: G \\ \\ \implies \sf \: Centripetal \: Force = Gravitational \: Force </p><p>\\ \\ \implies \sf \: \underline{\boxed{ \bold{ \blue{ \frac{m {V}^{2} }{(R + h)} = \frac{GMm}{ {(R + h)}^{2} }}}}} \\ \\ \implies \sf \: {V}^{2} = \frac{GM}{R + h} = \frac{GM}{r} \: ( \because \: R + h = r) \\ \\ \implies \sf \: \underline{\boxed{ \bold{ \orange{V{ \tiny{orbit}} = \sqrt{ \frac{GM}{r}} = \sqrt{gr}}}}} \\ \\ \implies \sf \: where \: g = gravitational \: acceleration</p><p>\: of \: planet \\ \\ \implies \sf \: \underline{ \boxed{ \bold{ \green{g = \frac{GM}{ {r}^{2} }}}}} \\ \\ \star \sf \: \underline{\boxed{ \bold{ \red{Facts \: :}}}} \\ \\ \implies \sf \: orbital \: velocity \: is \: not \: depends \: upon \: mass \: \\ \sf \: of \: satellite . \\ \\ \implies \sf \: value \: of \: orbital \: velocity \: for \: earth \: is \: 8 \: \frac{km}{s} \\ \\ \star \sf \:</p><p>\underline{\boxed{ \bold{ \red{Diagram \: :}}}} \\ \\ \implies \sf \: see \: attached \: image \: for \: better \: understanding.$