For a projectile to hit a target it normally must impact within an area which is the cross-sectional area of the target. For a spherical body, this area is r² where r is the radius of the sphere. However, in the case of a meteoroid or an asteroid impacting the Earth (or another planet), the appropriate cross- sectional radius is not the physical radius of the Earth, but it gravitational radius, which depends upon the impactor velocity.

This impact will be tangential if Rg is the gravitational radius of the Earth (for the space velocity v (ie the body just impacts the planet). We can use the laws of conservation of momentum aned conservation of energy to determine Rg.

v Rg = V Rp (conservation of momentum) v2/2 = V2/2 - G Me / Rp (conservation of energy)

where G is the universal gravitational constant. Solving these two equations simultaneously we find that:

Rg = Rp √[ 2 G Me / ( v2 Rp ) + 1 ]

The table below presents values of Rg (as a function of the space velocity), both in kilometres and as a fraction of the Earth's radius. Also shown are the fractional area, and the collision velocity.

Space vel(km/s)	Collision vel(km/s)	Effective radius(km)	Fractional radius	Fractional area
2 4 5 6 8 10 15 20 25 30 35 40 45 50	11 12 12 13 14 15 19 23 27 32 37 42 46 51	36237 18942 15630 13494 10965 9571 7958 7309 6989 6809 6698 6625 6574 6538	5.68 2.97 2.45 2.12 1.72 1.5 1.25 1.15 1.1 1.07 1.05 1.04 1.03 1.02	32.26 8.81 6 4.47 2.95 2.25 1.56 1.31 1.20 1.14 1.1 1.08 1.06 1.05

Material Prepared by John Kennewell. © Copyright IPS - Radio and Space Services.