How Do You Solve Collision Equations for Final Velocities?

[Super6]

Can anyone help show me the process of solving the equations
(m1*v1)+(M2+V2)=(m1+v1')+(M2+V2')
(.5m1*v1^2)+(.5M2*V2^2)=(.5m1*v1')+(.5M2*V2')

to get the equations

----m1+M2 2M2v
v1' ----- v1 + ----- V2
----m1+M2 m1+M2


-----2m1 M2-m1
V2' ----- v1 + ----- V2
----m1+M2 m1+M2


i need the algebra used or at least how i should start i know you solve one and plug it into the other, but I am not sure if i should start by factoring, expanding or what any help would be greatly appreciated. I started by factoring but just need a shove in the right direction.

 

[HallsofIvy]

One thing you need is practice in copying the problem correctly:
in some places you have + where you should have * and you are missing "squares" in the second equation.

I take it you are given v1 and V2 and need to solve for v1' and V2'.

It's actually a lot easier to solve specific problems (with actual numbers for m1, m2, v1, and V2) than to solve the general formulas.

I would recommend that you solve the first equation for v1':
m1*v1'= m1v1+m2V2-m2v2' so v1'= (m1v1+m2V2-m2v2')/m1

and substitute that into the second equation so that you have an equation for V2' only (involving v1 and V2, of course). Solve that for V2' then substitute back into v1'= (m1v1+m2V2-m2v2')/m1 for v1'.

(One obvious simplification is to multiply that second (kinetic energy) equation by 2!)

 

[M98Ranger]

Sure, I can help explain the process of solving these collision equations for you. First, let's go over the variables in these equations:

m1 = mass of object 1
v1 = initial velocity of object 1
M2 = mass of object 2
V2 = initial velocity of object 2
v1' = final velocity of object 1
V2' = final velocity of object 2

To solve these equations, we will use the principles of conservation of momentum and conservation of kinetic energy.

Conservation of momentum states that the total momentum of a system remains constant before and after a collision. Mathematically, this can be represented as:

m1*v1 + M2*V2 = m1*v1' + M2*V2'

This equation shows that the total momentum of object 1 before the collision (m1*v1) is equal to the total momentum of object 1 after the collision (m1*v1') plus the total momentum of object 2 after the collision (M2*V2').

Next, we will use the principle of conservation of kinetic energy, which states that the total kinetic energy of a system remains constant before and after a collision. Mathematically, this can be represented as:

(0.5*m1*v1^2) + (0.5*M2*V2^2) = (0.5*m1*v1'^2) + (0.5*M2*V2'^2)

This equation shows that the total kinetic energy of object 1 before the collision (0.5*m1*v1^2) is equal to the total kinetic energy of object 1 after the collision (0.5*m1*v1'^2) plus the total kinetic energy of object 2 after the collision (0.5*M2*V2'^2).

Now, we can solve these equations simultaneously to find the final velocities (v1' and V2') of the objects after the collision. Here is one way to solve them algebraically:

1. Rearrange the momentum equation to solve for v1':

v1' = (m1*v1 + M2*V2 - M2*V2')/m1

2. Substitute this value for v1' into the kinetic energy equation:

(0.5*m1*v1^2) + (0.5*M2*V2^2