Physics homework problem-Very confused, due tomorrow, please help:)

Question

Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 9.0 km and a mass of 1.0×10¹⁵ kg hits the earth with an impact speed of 4.0×10⁴ m/s.

1. What is the earth's recoil speed after such a collision? Assume that the mass of the earth is 5.98⋅10^24 kg. (Use a reference frame in which the earth was initially at rest.)
2. What percentage is this of the earth's speed around the sun? Assume that the orbit of the earth is a circle with the radius of 1.5⋅10^11 m.

Answer 1

Let

M = 5.97 × 10²⁴ kg be the mass of the Earth,

m = 1.0 × 10¹⁵ kg be the mass of the asteroid,

v = 4.0 × 10⁴ m/s be the impact speed of the asteroid,

w (to be computed) be the recoil velocity of the Earth,

R = 1.5×10¹¹ m be the distance between the Earth and the sun,

T = 1 year ≈ π×10⁷ s.

1.

Recoil, in general, is the consequence of conservation of momentum.

The momentum of the pair (Earth, asteroid) before the collision

must equal the final momentum of the Earth with the asteroid imbedded in it.

Before the collision the total momentum is

M×0 + m×v

After the collision the total momentum is

(M+m)× w

Setting those two equal

(M+m)w = mv

So

w = mv/(M+m) = 1.0 × 10¹⁵ × 4.0 × 10⁴ /(5.97 × 10²⁴ + 1.0 × 10¹⁵) = 6.7 × 10^-6 m/s

2.

The velocity of the Earth around the Sun is

2πR/T = 2×π×1.5×10¹¹/π×10⁷ = 3×10⁴ m/s

The ratio between the recoil velocity and the Earth's speed around the Sun is

6.7×10^-6 / 3×10⁴ = 2.2×10^-10 = 2.2×10^-8%