Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the mid-point of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

Question

Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the mid-point of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are) (A) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 4 √ (GM/L) (B) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2 √ (GM/L) (C) The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is √ (2GM/L) (D) The energy of the mass m remains constant

Tardigrade Admin · Accepted Answer

Let v is the minimum velocity. From energy conservation, Uc + Kc = U∞ + K∞ ∴ m Vc + (1/2) m v2 =0+0 ∴ v= √-2Vc = √(-2) ( (-2GM/L) ) = 2 √( GM/L )

Q. Two bodies, each of mass $M$ , are kept fixed with a separation $2 L$ . A particle of mass m is projected from the mid-point of the line joining their centres, perpendicular to the line. The gravitational constant is $G$ . The correct statement(s) is (are)

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $4 \frac{GM}{L}$

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $2 \frac{GM}{L}$

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is $\frac{2 GM}{L}$

The energy of the mass m remains constant

Let v is the minimum velocity. From energy conservation,
$U_{c} + K_{c} = U_{\infty} + K_{\infty}$
$∴ m V_{c} + \frac{1}{2} m v^{2} = 0 + 0$
$∴ v = - 2 V_{c} = (- 2) (\frac{- 2 GM}{L}) = 2 \frac{GM}{L}$

Q. Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the mid-point of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 4LGM​​

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is 2LGM​​

The minimum initial velocity of the mass m to escape the gravitational field of the two bodies is L2GM​​