Question

Two meteoroids are 2.0×10⁵ km from Earth's center and heading straight toward Earth. One is moving at 12 km/s , the other at 20 km/s . a. At what speed will the first meteoride strike the Earth? b. At what speed will the second meteoride strike the Earth?

Answer 1

The body-1 (1) and body-2 (2) initial (i) and final (f) velocities v1i, v2i, v1f, v2f;

specific kinetic energies (energy/mass) of the bodies SKE1i, SKE2i, SKE1f, SKE2f;

and also the change in specific potential energy ΔSPE which is the same for both bodies.

G = 6.67428 X 10-11 N-m^2/kg^2, M(earth) = 5.9742X10²⁴ kg.

SPE = -GM/r

Change in SPE from 2.0×105km to r(earth) 6378 km:
ΔSPE = -GM(1/6378000-1/2.0×10⁵⁾= 1931156958


The long version:
SKE1i = v1i^2/2
SKE2i = v2i^2/2
SKE1f = SKE1i-ΔSPE
SKE2f = SKE2i-ΔSPE
v1f = sqrt(2SKE1f)
v2f = sqrt(2SKE2f)
Now,
v1f = sqrt(v1i^2-2ΔSPE)
v2f = sqrt(v2i^2-2ΔSPE)
v1 =sqrt(12²-2X1931156958)) = 12831.88217m/s
v2 = sqrt(20²-2X1931156958) = 14258.5887m/s