I have a question regarding direction of momentum

Question

So a few homeworks ago I had this question:

 

A billiard player sends the cue ball toward a group of three balls that are initially at rest and in contact with one another. After the cue ball strikes the group, the four balls scatter, each traveling in a different direction with different speeds as shown in the figure below. If each ball has the same mass, 0.16 kg, determine the total momentum of the system consisting of the four balls immediately after the collision. (Assume v1 = 0.31 m/s, θ1 = 70°, v2 = 0.49 m/s, θ2 = 30°, v3 = 0.21 m/s, v4 = 0.48 m/s.)

 

It asked me to find the magnitude which I found to be .027 kg * m/s but then it asked me to find the direction of the magnitude counterclockwise from the x-axis. 

 

This wasn't the first question on that homework that asked me to find the direction of the magnitude. Could someone explain this to me a little more clearly than what I'm reading in the textbook or in the lecture notes? The physics class is online so I suppose I overlook things here and there. 

Answer 1

Since we have angles here, this is just like analyzing vectors.  Recall that when we find the resultant vector, we find the x and y components of the vectors.  We combine the x components and combine the y-components.  This will give us the resultant x and resultant y.  Then we use Pythagorean theorem on these resultants to obtain the resultant of the system.

 

R = √(R_x² + R_y²)

 

In addition, when we want to find the direction of the resultant vector, we divide the resultant y by the resultant x, and take the inverse tangent of the result.

 

θ_R = tan^-1(R_y / R_x)

 

 

Okay, now that we had the review on vectors, lets apply it to momentum.  We know that momentum is mass times velocity.

 

p = mv

 

where:

p = momentum

m = mass

v = speed

 

Applying the vector concept,

 

p_1x = mv₁cosθ₁

p_1y = mv₁sinθ₁

 

p_2x = mv₂cosθ₂

p_2y = mv₂sinθ₂

 

Do the same for the 3rd and 4th ball.

 

 

Combine their x and y components, then use Pythagorean theorem.