Asteroid collision

Question

A warning system has calculated that two asteroids will collide not far from Earth any time soon.

The smaller asteroid has the massmandmoves with the velocity vm. The bigger asteroid has the

massM = 3mand the velocity of vM = 1

2vm. They collide at an angle of = 60 and turn into a

single heavy asteroid (inelastic collision):

M

m

(a) Calculate the velocity of the single object aer the collision.

(b) Determine the angle aer the collision.

Answer 1

Conservation of momentum and some vector math

We look at the total momentum of the system before the collision and this must still be the momentum of the single object after the collision.

Determining the angle....relative to what?

I am going to assume relative to the velocity vector of the of the less massive asteroid.

By conservation of momentum the directional momenta are also conserved in this collision.

The tan(relative angle) is conserved and equal Total momentum y/Total momentum x both before and after the collision.

Setting up the coordinate system for solution- suggest we pick directional vector for smaller mass asteroid to coincide with our x axis

so tan( angle) = momentum larger mass x sin 60/(momentum small mass + momentum large mass x cos 60)

The momentum in the y direction (direction perpendicular to the momentum of the smaller mass asteroid) is conserved. That is momentum larger mass x sin 60.

Having solved for the angle after collision call it alpha in our first ratio.

We have the Momentum of the single heavy asteroid x sin alpha = momentum larger mass x sin 60.

and v final = momentum larger massx sin60/sin alpha / (M + m) M+m the mass of the single heavy asteroid.