A mass m,travelling at speed V_0 in a straigh | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) $1$. Conservation of angular momentum about the black hole:$mv_0 R = mva \implies v = \frac{v_0 R}{a}$$2$. Conservation of mechanical energy:Initi

Vedclass · Accepted Answer

$a = R\left(1 + \frac{2GM}{aV_0^2}\right)^{-1/2}$

Vedclass · Answer

(D) $1$. Conservation of angular momentum about the black hole:$mv_0 R = mva \implies v = \frac{v_0 R}{a}$$2$. Conservation of mechanical energy:Initial energy = Final energy at closest approach$\frac{1}{2}mv_0^2 + 0 = \frac{1}{2}mv^2 - \frac{GMm}{a}$$3$. Substitute $v$ into the energy equation:$\frac{1}{2}v_0^2 = \frac{1}{2}\left(\frac{v_0 R}{a}\right)^2 - \frac{GM}{a}$$v_0^2 = \frac{v_0^2 R^2}{a^2} - \frac{2GM}{a}$$1 = \frac{R^2}{a^2} - \frac{2GM}{a v_0^2}$$\frac{R^2}{a^2} = 1 + \frac{2GM}{a v_0^2}$$\frac{R}{a} = \left(1 + \frac{2GM}{a v_0^2}\right)^{1/2}$$a = R\left(1 + \frac{2GM}{a v_0^2}\right)^{-1/2}$

Similar Questions

$A$ study of binary stars is most helpful in

$A$ body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest

$Assertion$ : The Earth is slowing down and as a result the Moon is coming nearer to it.
$Reason$ : The angular momentum of the Earth-Moon system is not conserved.

$Assertion$ : The length of the day is slowly increasing.
$Reason$ : The dominant effect causing a slowdown in the rotation of the earth is the gravitational pull of other planets in the solar system.

Vedclass Test Series

Exam Paper Generator

Online Exam Module