A rocket initially moving with velocity v0 towards the moon...

A rocket, initially moving with velocity (v_{0}) towards the Moon of mass (M) and radius (r_{0}) is considered. Determine the cross section (sigma) for striking the Moon, assuming the Moon is at rest and disregarding other celestial bodies. Hint: (sigma) represents the area of the cross section of the cylinder formed by initially parallel orbits far from the Moon that eventually collide with it. The increase in this cross section over the purely geometric cross section (pi r_{0}^{2}) is referred to as gravitational focusing.

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00:01 Hello students in this question we are given a rocket which is moving so it is moving towards the moon and it is the moon is at rest that is the assumption so we need to find out the cross section sigma for striking the moon to solve this we need to calculate the cross section sigma so we can consider a cross sectional area cylinder like cylinder of cross sectional area sorry a right this is the cross sectional area and the length is equal to the distance traveled by the rocket right this is the length like length is equal to the distance traveled by the rocket before striking the moon so if the cylinder axis is parallel to the rocket's velocity vector this is the velocity vector and the axis of the cylinder is parallel to that okay then the volume of the cylinder is accepted by the rocket during the journey v is given by area times velocity that will be the volume total volume so due to moon's gravitational field the rocket's trajectory is bent inward right resulting in gravitational focusing this causes a cylinder cross sectional area to increase by a factor of f so this is known as gravitational focusing factor so effective cross sectional area sigma is equal to f times a to calculate f we can use the concept of impact parameter so consider a point of mass m at rest at the center of the moon suppose that the rocket approaches the moon along this straight line passing a distance b from the center of moon moon okay um moon then the minimum distance d is the closest approach right that is the closest approach so distance is equal to b minus r zero so where r is the radius and b is the distance from the surface so that is the closest approach which is called b minus r zero so the impact parameter b can take any value between zero to infinity it can go from zero to infinity you can go but only those trajectories with the impact parameter between b a maximum value and b minimum value can be adjusted accepted and that will result in the rocket striking this moon so b minimum will be equal to r zero plus r zero g m by v zero square the whole raised to one by three that will be the b minimum and b maximum will be equal to r zero plus 27 r zero square…

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