The masses and radii of the earth and moon are M1, R1 and M2, R2 respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is

Question

The masses and radii of the earth and moon are M1, R1 and M2, R2 respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is (A) √[2](G/a)(M1+M2) (B) √[2](2 G/a)(M1+M2) (C) √[2](G m/a)(M1+M2) (D) √[2](G m(M1+M2)/d(-R1+R2))

Tardigrade Admin · Accepted Answer

Condition for escaping of a particle is KE=PE or (1/2) m v2=(G M1 m/d / 2)+(G M2 m/d / 2) v2=(4 G/d)(M1+M2) v=2 √(G(M1+M2)/d)

Q. The masses and radii of the earth and moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centres so that it escapes to infinity is

$2 \frac{G}{a} (M_{1} + M_{2})$

$2 \frac{2 G}{a} (M_{1} + M_{2})$

$2 \frac{G m}{a} (M_{1} + M_{2})$

$2 \frac{G m ( M _{1} + M _{2} )}{d ( - R _{1} + R _{2} )}$

Condition for escaping of a particle is $K E = PE$
or $\frac{1}{2} m v^{2} = \frac{G M _{1} m}{d /2} + \frac{G M _{2} m}{d /2}$
$v^{2} = \frac{4 G}{d} (M_{1} + M_{2})$
$v = 2 \frac{G ( M _{1} + M _{2} )}{d}$

Q. The masses and radii of the earth and moon are M1​,R1​ and M2​,R2​ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is

2aG​(M1​+M2​)​

2a2G​(M1​+M2​)​

2aGm​(M1​+M2​)​

2d(−R1​+R2​)Gm(M1​+M2​)​​