What is the final velocity of two colliding masses?

[teggenspiller]

Homework Statement

two identical bodies are sliding toward each other on a frictionless surface. One moves at 1 m/s and the other at 2 m/s. They collide and stick. The magnitude of the velocity of the combined mass is

A. 3/4 m/s.
B. 2/3 m/s.
C. 1/6 m/s.
D. 1/2 m/s.
E. 1.5 m/s.
F. none of these

Homework Equations

m(v2-v1)=m(v2-v1)

The Attempt at a Solution

This is seriously confusing me. So in the beginning, the masses are irrelvent since they are the same and not mentioned.

the initial momentum of the system is 2+1, so 3?

and i was under the impression that the momentum conservation theory meant the momentum stayed the same, so the final momentum is also 3, correct?

 

[Fewmet]

Momentum is a vector. If the two bodies slide toward each other, one must be positive and the other negative.

That will give you one of the choices.

 

[Idoubt]

you consider the two masses a system, then the momentum of the system is conserved ( because there are no external forces ) you should determine what that is before and after collision

 

[Idoubt]

Homework Statement

the initial momentum of the system is 2+1, so 3?

and i was under the impression that the momentum conservation theory meant the momentum stayed the same, so the final momentum is also 3, correct?

sorry didn't read the last line, well you forgot the masses, and also the direction of the velocities ( just because the masses are the same doesn't mean momentum doesn't depend on it )

 

[Disconnected]

the initial momentum of the system is 2+1, so 3?

and i was under the impression that the momentum conservation theory meant the momentum stayed the same, so the final momentum is also 3, correct?

This is where you got mixed up...
Momentum is a vector, so it has direction.

p=mv (bold denotes vectors)

So the direction of the momentum of an object is the same as the direction of that object's velocity, right?

So let's take the direction of object 1 to be positive, and thus the direction of object two to be negative. The masses will clearly all cancel at the end, so I'll just call them 1.

p1=1*1

p2=1*(-2)

Thus the initial momentum is 1+(-2)=-1

Thus the final momentum is -1.

p=mv

v=p/m

v=(-1)/2
|v=|=1/2

Make sense?

 

[teggenspiller]

gosh duh. THANK YOU so much disconnected. so how do you know when to subtract and when to add velocities, do i just have to analyze their directions?

for example, in
Two trucks with the same masses are moving toward each other along a straight line with speeds of 50 mi/h and 60 mi/h. What is the speed of combined trucks after completely inelastic collision?
i assume the 50m/s is moving east, while 60m/s mass is moving west. That would make their final combined velocity

pi=1*50
p2= 1*(-60)
50+(-60)= -1 <--i put this negative one here, cause that's what you did.
10mi/h
^^^^which is actually incorrect..


by the way, why is this 1 suddenly negative in the explantaion you gave me.. i just ran into this reading over it;
"p2=1*(-2)

Thus the initial momentum is 1+(-2)=-1

Thus the final momentum is -1."

 

[Fewmet]

gosh duh. THANK YOU so much disconnected. so how do you know when to subtract and when to add velocities, do i just have to analyze their directions?

for example, in
Two trucks with the same masses are moving toward each other along a straight line with speeds of 50 mi/h and 60 mi/h. What is the speed of combined trucks after completely inelastic collision?
i assume the 50m/s is moving east, while 60m/s mass is moving west. That would make their final combined velocity...

I encourage my students to start every problem with a sketch that shows one way as a positive and the other as negative. Suppose, in your example with the trucks, east is positive and west is negative. That means any velocity, acceleration, force. momentum or displacement that is directed westward is negative. Any directed eastward is positive. That gives you
50M + -60M = -10M
The fact that the total momentum is negative tell you that it is directed eastward.

The same works for up and down needing to have opposite signs.

 

[Disconnected]

gosh duh. THANK YOU so much disconnected. so how do you know when to subtract and when to add velocities, do i just have to analyze their directions?

Yeah. Like fewmet said, it's best to draw a picture (initially on paper, but eventually in your minds eye) and designate one of the directions as positive (the direction you choose is arbatrary, you are defining the coordinates used) and the other, obviously, as negative. Things get a little more complex as you add dimensions, but the same principles hold.

The combination of vectors is always addition, however one or more may have a negative value.

by the way, why is this 1 suddenly negative in the explantaion you gave me.. i just ran into this reading over it;
"p2=1*(-2)

Thus the initial momentum is 1+(-2)=-1

Thus the final momentum is -1."

In my notation p1 was the momentum of particle 1 and p2 was the momentum of particle 2.
So the momentum is give by the mass times the velocity, and the velocity (a vector!) of particle 2 is 2m/s in the negative direction, and is thus -2. 1*(-2)=-2, right? so the momentum of particle two is of magnitude 2 but in the negative direction, thus -2.

The momentum of particle 1 is just 1*1 because it's velocity is one in the positive direction=1.

So the total momentum is p1+p2, right? so 1+(-2) gives the result of -1, agreed? So the total momentum is magnitude 1, but pointed in the negative direction. This makes sense as the objects are of equal mass and the one moving in the negative direction is going faster.

The 1 isn't suddenly negative, it's just the result of adding 1 and -2!

So the initial momentum is -1, thus the final momentum is -1.

Remember that the new mass is the sum of the two initial masses, which were set to 1. Therefore the new mass is 2. We will call the momentum of the two objects moving together p12.

p12=2*v

But we know that p12=-1 from conservation of momentum, thus

-1=2*v

thus

v=-1/2

So the resulting velocity is magnitude 0.5, and in the negative direction - which makes sense. If you threw two balls of putty at each other and one was going faster then the other (and the masses were equal), you would expect the resulting motion to be in the same direction as the one that was traveling faster (thrown harder), right?

As the question asks for the magnitude |p|, the negative sign disappears.

Notice that if you define the coordinates as v1 being negative and v2 being positive, the final momentum will be positive 0.5. Again this makes sense as this new coordinate system is the inverse of the initial system, so any value that used to be negative is now positive and visa versa.

Two trucks with the same masses are moving toward each other along a straight line with speeds of 50 mi/h and 60 mi/h. What is the speed of combined trucks after completely inelastic collision?
i assume the 50m/s is moving east, while 60m/s mass is moving west. That would make their final combined velocity

pi=1*50
p2= 1*(-60)
50+(-60)= -1 <--i put this negative one here, cause that's what you did.
10mi/h
^^^^which is actually incorrect..

Eeeep! 50+(-60)=-10, not -1!

And remember that the final mass is the sum of the masses, which in this case is again 2.

so
p12=2*v

-10=2*v

v=-5

Make sense? Feel free to ask if I didn't explain it right. I am very prone to silly little mistakes (like forgetting that gravity exists...)!