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Distance between the centres of two stars is 10a. The masses of the 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 to the smaller one. What would be the minimum initial speed to reach the smaller star?

Distance between the centres of two stars is 10a. The masses of the 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 to the smaller one. What would be the minimum initial speed to reach the smaller star?

Grade:11

1 Answers

Arun
25750 Points
5 years ago
The object needs enough energy to make it to the position where the gravitational forces from the two stars are equal. Beyond that, it will accelerate towards the smaller star. 

If the distance from the larger star's center is x, then this position is: 
GMm/(10a-x)^2 = G(16M)m/x^2 
x = 8a 

At its starting position, the gravitational potential energies with respect to the two stars are -GMm/(8a) and -G(16M)m/(2a). 
At the balanced position, the gravitational potential energies with respect to the two stars are -GMm/(2a) and -G(16M)m/(8a). 
Moving the object has increased its total potential energy by (45/8)GMm/a. 

Therefore it needs to start with kinetic energy of at least (45/8)GMm/a. 
mv^2/2 = (45/8)GMm/a 
v = 3/2 sqrt(5GM/a)

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