Question

Show that the escope velocity is given by √gr

Answer 1

Answer:

Escape Velocity is √2 times of orbital Velocity.

Proof:

Orbital velocity of satellites is given as :

\begin{gathered}orbital \: v = \sqrt{ \frac{Gm}{r} } \\ \end{gathered}

orbitalv=

r

Gm

Where m => mass of Earth , r => radius of Earth.

Escape Velocity can be calculated by the us of Conservation of Energy theorem:

KE1 + PE1 = KE2 + PE2

\frac{1}{2} M{(v. \: esc)}^{2} - \frac{GmM}{ {r}^{2} } = 0 + 0

2

1

M(v.esc)

2

−

r

2

GmM

=0+0

\begin{gathered} = > v \: esc. = \sqrt{ \frac{2Gm}{r} } \\ \end{gathered}

=>vesc.=

r

2Gm

\begin{gathered} = > v \: esc. = \sqrt{2 } \times ( \sqrt{ \frac{Gm}{r} } ) \\ \end{gathered}

=>vesc.=

2

×(

r

Gm

)

= > v \: esc. = \sqrt{2} \times (orbital \: v)=>vesc.=

2

×(orbitalv)

[Hence proved.