## Question

The masses and radii of the earth and moon are *M*_{1}, *R*_{1} and *M*_{2}, *R*_{2}respectively. Their centres are at distance *d* apart. What is the minimum speed with which a particle of mass *m* should be projected from a point midway between the two centres so as to escape to infinity?

### Solution

Potential energy of *m* when it is midway between *M*_{1} and *M*_{2},

And as potential energy at infinity is zero, so work required to shift *m* from the given position to infinity,

As this work is provided by initial kinetic energy,

#### SIMILAR QUESTIONS

Imagine a planet whose diameter and mass are both one-half of those of earth. The day’s surface temperature of this planet reaches upto 800 K. Are oxygen molecules possible in the atmosphere of this planet? Give calculation. (Escape velocity on earth’s surface = 11.2 km s^{–1}, Boltzmann’s constant

*k* = 1.38 × 10^{–23} JK^{–1}, mass of oxygen molecule = 5.3 × 10^{–26 }kg.)

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Distance between the centres of two stars is 10*a*. The masses of these stars are *M* and 16*M* and their radii *a* and 2*a*, 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*.