An asteroid of mass m is approaching earth, initially at a distance of 10 Re with speed vi. It hits the earth with a speed vf(Re. and Me are radius and mass of earth), then

Question

An asteroid of mass m is approaching earth, initially at a distance of 10 Re with speed vi. It hits the earth with a speed vf(Re. and Me are radius and mass of earth), then (A) vf2=vi2+(2 G m/Me R)(1-(1/10)) (B) vf2=vi2+(2 G Me/Re)(1+(1/10)) (C) vf2=vi2+(2 G Me/Re)(1-(1/10)) (D) vf2=vi2+(2 G m/Re)(1-(1/10))

Tardigrade Admin · Accepted Answer

Applying law of conservation of energy for asteroid at a distance

10 R_{e}

and at earth's surface,

K_{i} + U_{i} = K_{f} + U_{f}

(i)
Now,

K_{i} = \frac{1}{2} m v_{i}^{2}

and

U_{i} = \frac{G M _{e} m}{10 R _{e}}

K_{f} = \frac{1}{2} m v_{f}^{2} and U_{f} = - \frac{G M _{e} m}{R _{e}}

Substituting these values in Eq. (i), we get

\frac{1}{2} m v_{i}^{2} - \frac{G M _{e} m}{10 R _{e}} = \frac{1}{2} m v_{f}^{2} - \frac{G M _{e} m}{R _{e}}

\Rightarrow \frac{1}{2} m v_{f}^{2} = \frac{1}{2} m v_{i}^{2} + \frac{G M _{e} m}{R _{e}} - \frac{G M _{e} m}{10 R _{e}}

\Rightarrow v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} - \frac{2 G M _{e}}{10 R _{e}}

∴ v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} (1 - \frac{1}{10})

Q. An asteroid of mass $m$ is approaching earth, initially at a distance of $10 R_{e}$ with speed $v_{i}$ . It hits the earth with a speed $v_{f} (R_{e}$ and $M_{e}$ are radius and mass of earth), then

$v_{f}^{2} = v_{i}^{2} + \frac{2 G m}{M _{e} R} (1 - \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} (1 + \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} (1 - \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G m}{R _{e}} (1 - \frac{1}{10})$

Q. An asteroid of mass m is approaching earth, initially at a distance of 10Re​ with speed vi​. It hits the earth with a speed vf​(Re​ and Me​ are radius and mass of earth), then

vf2​=vi2​+Me​R2Gm​(1−101​)

vf2​=vi2​+Re​2GMe​​(1+101​)

vf2​=vi2​+Re​2GMe​​(1−101​)

vf2​=vi2​+Re​2Gm​(1−101​)

Q. An asteroid of mass $m$ is approaching earth, initially at a distance of $10 R_{e}$ with speed $v_{i}$ . It hits the earth with a speed $v_{f} (R_{e}$ and $M_{e}$ are radius and mass of earth), then

$v_{f}^{2} = v_{i}^{2} + \frac{2 G m}{M _{e} R} (1 - \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} (1 + \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G M _{e}}{R _{e}} (1 - \frac{1}{10})$

$v_{f}^{2} = v_{i}^{2} + \frac{2 G m}{R _{e}} (1 - \frac{1}{10})$