Distance between the centres 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 larger star towards the smaller star- What should be its minimum initial speed to reach the surface of the smaller star- Obtain the expression in terms of G- M and a-

Question

Distance between the centres 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 larger star towards the smaller star- What should be its minimum initial speed to reach the surface of the smaller star- Obtain the expression in terms of G- M and a-

Toppr 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,
$\frac{G M}{r_{1}^{2}} = \frac{G (16 M)}{r_{2}^{2}}$
or $\frac{r_{2}}{r_{1}} = 4$
also $r_{1} + r_{2} = 10 a$
$∴$ $r_{2} = (\frac{4}{4 + 1}) (10 a) = 8 a$
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 $\frac{1}{2} m v_{m i n}^{2}$
=Potential energy of the body at P-Potential energy at the surface of the larger star.
$∴$ $\frac{1}{2} m v_{m i n}^{2} = [- \frac{G M m}{r_{1}} - \frac{16 G M m}{r_{2}}] - [- \frac{G M m}{10 a - 2 a} - \frac{16 G M m}{2 a}]$
$= [- \frac{G M m}{2 a} - \frac{16 G M m}{8 a}] - [- \frac{G M m}{8 a} - \frac{8 G M m}{a}]$
or $\frac{1}{2} m v_{m i n}^{2} = (\frac{45}{8}) \frac{G M m}{a}$
$∴$ $v_{m i n} = \frac{3 \sqrt{5}}{2} (\sqrt{\frac{G M}{a}})$