Dot product issue. system of equations

[bcddd214]

Homework Statement

As illustrated, a ball of mass m_1=0.25 kg and velocity V_(0_1=+5.00 m/s) collides head on with a ball of mass m_2=0.8 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after they collide?

Homework Equations

V_(0_1 )
p_i=p_f
〖KE〗_i=〖KE〗_f

The Attempt at a Solution

m_1 v_(1_i )+0=m_2 v_(2_f )+m_1 v_(1_f )
1/2 m_1 〖v_(1_i )〗^2=1/2 m_1 〖v_(1_f )〗^2+1/2 m_2 〖v_(2_f )〗^2
A ⃑*B ⃑=■(i ̂&j ̂&k ̂@A_x&A_y&A_z@B_x&B_y&B_z )=|■(A_y&A_z@B_y&B_x )| i ̂+
I get a total brain fart once I plug it into the matrix...? :(

 

[ideasrule]

A ⃑*B ⃑=■(i ̂&j ̂&k ̂@A_x&A_y&A_z@B_x&B_y&B_z )=|■(A_y&A_z@B_y&B_x )| i ̂+
I get a total brain fart once I plug it into the matrix...? :(

That last line doesn't seem to render well on my computer, but why do you need to plug anything into a matrix? You have:
m_1 v_(1_i )+0=m_2 v_(2_f )+m_1 v_(1_f )

and:

1/2 m_1 〖v_(1_i )〗^2=1/2 m_1 〖v_(1_f )〗^2+1/2 m_2 〖v_(2_f )〗^2

You know v_1_i, and you just need to solve for either v_1_f or v_2_f. Expressing one of them in terms of the other using the first equation, then plugging the result into the second, should give you the result.

 

[bcddd214]

I am sooo physics paranoid. I hit the table and freeze because I keep getting it wrong for some reason.

A*B=A_x, A_y, A_z= |A_y, A_z|i
_____B_x, B_y, B_z |B_y, B_x|

I get here, I seem to be on a roll, and then just freeze. :(

 

[ideasrule]

I still don't understand why you're using matricies. What's A and B? Why are you multiplying them together?

You already got 2 equations, and you have only 2 unknowns. Direct substitution should be all you need to get the answer. If you're trying to use Gaussian elimination, it doesn't work because this isn't a linear system of equations.

 

[rwisz]

The dot product issue that you are facing is a common problem in solving systems of equations, especially when dealing with vectors. The dot product is used to find the angle between two vectors, but in your problem, you are trying to solve for the velocities of the two balls after the collision.

One possible approach to solving this problem is to use the conservation of momentum and the conservation of kinetic energy equations. The first equation states that the initial momentum (p_i) of the system is equal to the final momentum (p_f) of the system. In this case, the initial momentum is equal to the mass of the first ball times its initial velocity, as the second ball is initially at rest. The final momentum is equal to the mass of the first ball times its final velocity, plus the mass of the second ball times its final velocity.

The second equation states that the initial kinetic energy (KE_i) of the system is equal to the final kinetic energy (KE_f) of the system. In this case, the initial kinetic energy is equal to the kinetic energy of the first ball, which is equal to 1/2 times its mass times the square of its initial velocity. The final kinetic energy is equal to the kinetic energy of the first ball, which is equal to 1/2 times its mass times the square of its final velocity, plus the kinetic energy of the second ball, which is equal to 1/2 times its mass times the square of its final velocity.

By setting these two equations equal to each other and solving for the final velocities of the two balls, you can find the solution to this problem without using the dot product. So, don't worry if you get stuck when trying to use the dot product, as there are other ways to solve this problem. Keep practicing and keep exploring different approaches to problems, and you'll become more comfortable with using the dot product in the future.