How Is the Equation a Dot Product in Momentum Problems?

[mohdhm]

Homework Statement

Ok so i ran into trouble in the momentum section because i do not understand dot products as well as i thought. I tried going back and revising my notes but nothing new comes to mind. Your help is highly appreciated.

ok so let me just state that m1=m2

the problem consists of 2 billiard balls, one is at rest and the other strikes it and sends it towards the corner pocket, they both share the same mass. the purpose is to find theta, but that is not what I'm trying to find out here.

We write the kinetic formula which gets reduced to v1i^2 = v1f^2 + v2f^2
then the momentum formula also gets reduced, this time it gets reduced to : v1i = v1f + v2f.

what i can't figure out, is that the example tells me to to square both sides (of the previous formula) and find the dot product.

then i get v1i^2 = (v1f + v2f)(v1f+v2f)... which gets expanded.. and so on
[the formula makes sense from a logical point of view]

The point is, how is the equation above, the DOT PRODUCT. I don't get that. i thought the dot product formula is AB = ABCOS(THETA)

any explanations?

 

[Mindscrape]

Dot product? I have absolutely no idea. The equation is simply a product. You can't just take dot products for no reason, like your book seems to have done to suddenly get an angle. Are you sure your book didn't make momentum vectors?

My advice is to do what makes sense to you. If your book uses some clever way to find the angle that the balls go off at, but you have a way that simply does it by looking at conservation of momentum in the x and y directions then you should do it your way.

Could you write out exactly what your book has done, or is this it? Dot products, in case you are confused, are merely a way to multiply two vectors. You can either multiply the like components, (i.e. A1x + A2y + A3z dot B1x + B2Y + B3Z = A1B1x + A2B2y + A3B3z), or you can use the formula you listed which is A dot B = ABcosØ.

 

[mohdhm]

i guess your right, this example in the book isn't even useful anyway, it is only used to determine the angle, and we can do that by this formula Phi + theta = 90 degrees. (only when the collision [k is conserved] is elastic and we have m1=m2)

 

[Hurkyl]

The dot product satisfies some properties. For example, it is distributive (just like ordinary multiplication)...

 

[Dick]

I think the point is that the formula "Phi + theta=90 degrees" can be derived by taking the dot product of the vector equation v1i=v1f+v2f with itself and applying energy conservation, the distributive law of which Hurkyl spoke and your A.B=|A||B|cos(phi).

 

[Mindscrape]

Yes, I imagine it was really doing something similar to using vectors and the dot product for proving law of cosines. Still, with what the poster wrote it isn't exactly a dot product. Given the velocity vectors, which I actually made a mistake earlier on thinking he was writing out the components in the i(hat) direction (I was tired), it should go more like:

$$\mathbf{v_1_0} = \mathbf{v_1_f} + \mathbf{v_2_f}$$

then square both sides of the formula

$$\mathbf{v_1_0}^2 = (\mathbf{v_1_f} + \mathbf{v_2_f})^2$$

which would be the vectors dotted with themselves

$$v{_1_0}^2 = (\mathbf{v_1_f} + \mathbf{v_2_f}) \cdot (\mathbf{v_1_f} + \mathbf{v_2_f})$$

then use the cosine distribution and dot products

$$v{_1_0}^2 = |v_1_f|^2+|v_2_f|^2 + 2*v_1_f*v_2_f*cos \theta$$

Still, it's obviously not something the book explained well, nor something I would expect an introductory physics course to go over and expect the students to use.

 

[mohdhm]

thanks for your contributions everyone.

 

[Dick]

Maybe not explained well, but it works. Either the incoming ball stops dead or the ricochet angle is 90 degrees. It's an interesting use of the dot product.