The Earth has an angular speed of 7.272 10-5 rad/s in its rotation. Find the new

Question

The Earth has an angular speed of 7.272 10-5 rad/s in its rotation. Find the new angular speed if an asteroid (m = 1.10 1022 kg) hits the Earth while traveling at a speed of 1.53 103 m/s (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases.
(a) The asteroid hits the Earth dead center along the radial line from the Sun through the Earth's center.


(b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation.


(c) The asteroid hits the Earth nearly tangentially in the direction opposite to Earth's rotation.

Explanation / Answer

The earth revolves at a rate of one revolution a day or 2.pi/86400 rad/sec.
When an asteroid hits the earth dead center, the energy of the asteroid will be transferred to the earth an they both will be traveling with the same speed. In case of an elastic collision both impulse and energy will be conserved. This can not be the case here, if that would be so the combined laws would give as result that the asteroid will be bouncing back. So the rule is that the impulse will be conserved, the energy also, but there will be energy converted to heat (explosion!). In this situation the earth will be (very slightly moving faster, but it would have no effect on the rotation.
If the asteroid hits the earth tangentially, the impulse will be divided over an impulse (vector) working on the center of the earth plus half the impulse vector on the place where the impact is, plus half the impulse vector on the opposite side of the earth. The two vectors on each side of the earth will add to the earth's rotation, if the hit is in the direction of the rotation, otherwise it will be subtracted. As in the first case: the impulse on the center of the earth will give the earth a slight push to a greater velocity.

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