Deriving Final Equation from Equations 1 & 2

[caligyrl4lyfe]

HI I need help with the steps in between? I am totally confused...

So, how to derive the final equation from equation 1 and 2


m1v1i+m2v2i=m1v1+m2v2f (Equation 1)

1/2m1V1i^2+1/2m2v2i^2=1/2m1v1f^2+1/2m2v2f^2 (Equation 2)

v1i-v2i= -(V1f-v2f) ( Final Equation)

 

[Trenthan]

ok it does work out just abit of algebra

First off what i did was rearrange your 2nd equation in terms of m1

Substitute m1 into your first equation and simplify down

you will end up with something like v_1f(v_2f^2-v_2i^2)+v_2i(v_1i^2-v_1f^2)=something

now you have to realize that (v_2f^2 - v_2i^2) = (v_2f - v_2i)(v_2f + v_2i)

More simplification from there and than you are done

(Tip don't expand all the brackets out it will drive you nuts, rather cancel them down after recognizing that "now you have to realize that (v_2f^2 - v_2i^2) = (v_2f - v_2i)(v_2f + v_2i)"

Thus from that you should get what are looking for

Cheers Trent

 

[thomate1]

To derive the final equation, we need to start by rearranging Equation 1 and 2 to solve for the final velocities (v1f and v2f).

For Equation 1, we can rearrange it as:

m1v1f = m1v1i + m2v2i - m2v2f

Then, we can substitute this into Equation 2 for v1f:

1/2m1V1i^2+1/2m2v2i^2=1/2(m1v1i + m2v2i - m2v2f)^2+1/2m2v2f^2

Expanding and simplifying, we get:

1/2m1V1i^2+1/2m2v2i^2=1/2(m1^2v1i^2 + 2m1m2v1iv2i + m2^2v2i^2 - 2m1m2v1if + m2^2v2f^2) + 1/2m2v2f^2

We can then combine like terms and solve for v2f:

1/2m1V1i^2+1/2m2v2i^2=1/2(m1^2v1i^2 + 2m1m2v1iv2i + m2^2v2i^2 + m2^2v2f^2) - m1m2v1if

1/2m1V1i^2+1/2m2v2i^2=1/2(m1^2v1i^2 + m2^2v2i^2 + 2m1m2v1iv2f + m2^2v2f^2)

m1m2v1if = 1/2m1V1i^2 + 1/2m2v2i^2 - 1/2(m1^2v1i^2 + m2^2v2i^2 + 2m1m2v1iv2f + m2^2v2f^2)

v1if = (1/2m1V1i^2 + 1