Distance between the center of two stars is 10a. the masses of these stars are m and 16M and their radii a and 2a respectively. A body of mass m is fired straight from the surface of the smller star towards the surface of the smaller star. What should be its minimum initial speed to reach the surface of the smaller star ?

Question

Distance between the center of two stars is 10a. the masses of these stars are m and 16M and their radii a and 2a respectively. A body of mass m is fired straight from the surface of the smller star towards the surface of the smaller star. What should be its minimum initial speed to reach the surface of the smaller star ? (A) √(GM/a) (B) (1/2)√(5GM/a) (C) (3/2)√(GM/a) (D) (3√5/2)√(GM/a)

Tardigrade Admin · Accepted Answer

Let there are two stars 1 and 2 as shown in figure
Let

P

is a point between

C_{1}

and

C_{2}

, where gravitational field strength is zero. Or at P field strength due to star 1 is equal and opposite to the field strength due to star 2 . Hence,

< b r / > \frac{GM}{r _{1}^{2}} = \frac{G ( 16 M )}{r _{2}^{2}} < b r / >

or

\frac{r _{2}}{r _{1}} = 4

also

r_{1} + r_{2} = 10 a

< b r / > ∴ r_{2} = (\frac{4}{4 + 1}) (10 a) = 8 a < b r / >

and

r_{1} = 2 a

Now, the body of mass

m

is projected from the surface of larger star towards the smaller one. Between

C_{2}

and

P

it is attracted towards 2 and between

C_{1}

and P it will be attracted towards 1 . Therefore, the body should be projected to just cross point

P

because beyond that the particle is attracted towards the smaller star itself.
From conservation of mechanical energy

\frac{1}{2} m v_{min}^{2}

=

Potential energy of the body at P-Potential energy at the surface of the larger star.

< b r / > < b r / > ∴ \frac{1}{2} m v_{m i n}^{2} = [- \frac{GM m}{r _{1}} - \frac{16 GM m}{r _{2}}] - [- \frac{GM m}{10 a - 2 a} - \frac{16 GM m}{2 a}] < b r / >= [- \frac{GM m}{2 a} - \frac{16 GM m}{8 a}] - [- \frac{GM m}{8 a} - \frac{8 GM m}{a}] < b r / > < b r / >

or

\frac{1}{2} m v_{m i n}^{2} = (\frac{45}{8}) \frac{GM m}{a}

∴ v_{m i n} = \frac{3 5}{2} (\frac{GM}{a})

$\frac{GM}{a}$

$\frac{1}{2} \frac{5 GM}{a}$

$\frac{3}{2} \frac{GM}{a}$

$\frac{3 5}{2} \frac{GM}{a}$

aGM​​

21​a5GM​​

23​aGM​​

235​​aGM​​

$\frac{GM}{a}$

$\frac{1}{2} \frac{5 GM}{a}$

$\frac{3}{2} \frac{GM}{a}$

$\frac{3 5}{2} \frac{GM}{a}$