Help!? Can someone show me how to do this problem?

Question

A billiard ball of mass mA=0.400kg moving with speed vA=180m/s strikes a second ball, initially at rest, of mss mB=0.400kg. As a result of the collision, the first ball is deflected off at an angle of 30.0° with a speed v'A=1.10m/s.

(a) Taking the x axis to be the original direction of motion of ball A, write down the equations expressing the conservation of momentum for the components in the x and y directions separately.

(b) Solve these equations for the speed v'B, and angle θ, of ball B. Do not assume the collision is elastic.

Answer 1

Well, although its unfortunate we can't treat this as an elastic collision, you're still in luck for they gave us the deflection angle.

Recall that in any collision momentum is conserved. Since momentum is mass*velocity, the initial momentum = mA*vA_initial + mB*vB_initial = mA*vA = 72 kg*m/s (since vB_initial is 0). To make our calculations more simple, we'll define our x and y axis such that vA is initially traveling parallel to the x-axis, and our y-axis is naturally perpendicular to our x-axis (as specified in the problem).

With that the final velocity of A in the x direction is vA_final * cos(30 deg), and the final velocity of A in the y direction is vA_final * sin(30 deg).

If you have trouble remembering when to use sine and when to use cosine, you can always draw a triangle, or consider that if the deflection is 30 degrees then most of the velocity is still in the x direction. Since cos(30) > sin(30) you know that cosine is in the x direction.

Now, the final momentum must equal the initial momentum. Since initially we had no momentum in the y direction, the sum of the final momentums in the y direction must also equal zero. Thus, the final momentum of B in the y direction is equal but opposite the final momentum of A in the y direction:
mB*vB_final * sin(θ) = mA*vA_final * sin(30) where their deflections are in different directions.
Likewise, the sum of the final momentums of A and B in the x direction must be equal to the original momentum of A in the x direction:
mA*vA_initial = mB*vB_final * cos(θ) + mA*vA_final * cos(30 deg).

Notice that since mA = mB, you can cancel all of the masses, comparing just the velocities.

You now have two equations with two unknowns, which you can solve with a little algebra.

For further practice, you may want to consider the initial and final kinetic energies to determine how much energy was absorbed in the collision.

I hope this helps!