Help with simple momentum derivation

[bsin]

Hi, I read this equation under elastic collisions in Wikipedia but I can't figure out how they derived it.

Homework Statement

For an elastic collision of two objects, show that the final velocity V1 is given by
V1 = [U1*(m1-m2) + 2*m2*U2]/[m1+m2]

where U1 = initial velocity of object1, U2 = initial velocity of object2 (known)
m1 = mass of object1 , m2 = mass of object2 (known)

Homework Equations

m1*U1 + m2*U2 = m1*V1 + m2*V2 (conservation of momentum)

[m1*U1^2]/2 + [m2*U2^2]/2 = [m1*V1^2]/2 + [m2*V2^2]/2 (conservation of energy)

The Attempt at a Solution

On Wiki, it said to change the frame of reference to make one of the unknown velocity, V1 or V2, equal to zero and solve for the other unknown velocity in the two conservation equations. I made V1 zero and juggled with a lot of algebra using substitution but couldn't arrive at anything similar to the above equation.

 

[PhanthomJay]

Hi, I read this equation under elastic collisions in Wikipedia but I can't figure out how they derived it.

Homework Statement

For an elastic collision of two objects, show that the final velocity V1 is given by
V1 = [U1*(m1-m2) + 2*m2*U2]/[m1+m2]

where U1 = initial velocity of object1, U2 = initial velocity of object2 (known)
m1 = mass of object1 , m2 = mass of object2 (known)

Homework Equations

m1*U1 + m2*U2 = m1*V1 + m2*V2 (conservation of momentum)

[m1*U1^2]/2 + [m2*U2^2]/2 = [m1*V1^2]/2 + [m2*V2^2]/2 (conservation of energy)

The Attempt at a Solution

On Wiki, it said to change the frame of reference to make one of the unknown velocity, V1 or V2, equal to zero and solve for the other unknown velocity in the two conservation equations. I made V1 zero and juggled with a lot of algebra using substitution but couldn't arrive at anything similar to the above equation.

You have 2 equations with 2 unknowns, so you have a solvable solution.
But it is not that simple to solve the 2 simultaneous equations. Just keep on surfing around for an explanation you are comfortable with. It's 'just' algebra.

 

[gneill]

Hi, I read this equation under elastic collisions in Wikipedia but I can't figure out how they derived it.

Homework Statement

For an elastic collision of two objects, show that the final velocity V1 is given by
V1 = [U1*(m1-m2) + 2*m2*U2]/[m1+m2]

where U1 = initial velocity of object1, U2 = initial velocity of object2 (known)
m1 = mass of object1 , m2 = mass of object2 (known)

Homework Equations

m1*U1 + m2*U2 = m1*V1 + m2*V2 (conservation of momentum)

[m1*U1^2]/2 + [m2*U2^2]/2 = [m1*V1^2]/2 + [m2*V2^2]/2 (conservation of energy)

The Attempt at a Solution

On Wiki, it said to change the frame of reference to make one of the unknown velocity, V1 or V2, equal to zero and solve for the other unknown velocity in the two conservation equations. I made V1 zero and juggled with a lot of algebra using substitution but couldn't arrive at anything similar to the above equation.

I find that it is easier to change the frame of reference to the center of momentum frame. In the COM frame, the collision is between two objects carrying equal and opposite momentum. This means that both objects "bounce", simply reversing their velocities.

So the procedure is:

1. Convert the initial velocities to the center of momentum frame.
2. Negate the resulting velocities.
3. Convert back to the original frame of reference.

Step 1 amounts to finding a velocity u to add to each initial velocity so that

(U1 + u)*m1 = -(U2 + u)*m2

Solve for u: u = -(m1*U1 + m2*U2)/(m1 + m2)

Step 2 says negate the resulting velocities:

v1 = -(U1 + u) and v2 = -(U2 + u)

Step 3 says convert back to original frame of reference:

V1 = v1 - u and V2 = v2 - u

Putting steps 2 & 3 together:

V1 = -(U1 + 2*u) and V2 = -(U2 + 2u)

I think that if you substitute the value for u from above into these expressions that you'll find the desired derived expressions.

 

[madah12]

I am not sure if it is accurate but I think of it like think assuming there is change in velocity
v₁m₁+u₁m₂=v₂m₁+u₂m₂


v₁m₁-v₂m₁=u₂m₂-u₁m₂
m₁(v₁-v₂)= m₂(u₂-u₁)

.5m₁v²₁+.5m₂u²₁=.5m₁v²₂+.5m₂u²₂

m₁v²₁+.m₂u²₁=m₁v²₂+m₂u²₂

then solve the two equations