Why does the equation for elastic collisions work like this?

[minnyveller]

Hi guys. Feels dumb coming back to this but I seem to have confused myself

I was helping a friend with the problem:

You shoot protons going v=4.2e7 m/s through a particle accelerator. They collide with gas particles of an unknown mass (pretend no velocity) and all bounce back elastically at 3.9e7 m/s.

I went through the steps of deriving the equation for m2 using the two speeds as follows:

m1v1^2 = m1v1'^2 + m2v2^2
m1v1 = m1v1' + m2v2

m1v1^2 - m1v1'^2 = m2v2^2
m1v1 - m1v1' = m2v2
v1 - v1' = v2

plugging this in ^

m1v1^2 = m1v1'^2 + m2(v1-v1')^2
m1(v1^2 - v1'^2) = m2(v1-v1')^2

m1(v1^2 - v1'^2)/(v1-v1')^2 = m2

plugging in both v1 and v1' as positive values (4.2e7 and 3.9e7) you get m2=27m (which was correct)

why does it only work plugging both in as positive values? Since the proton bounces back, if 4.2e7 is positive, why is the correct answer given when 3.9e7 is put in positive?

 

[Orodruin]

m1v1^2 - m1v1'^2 = m2v2^2
m1v1 - m1v1' = m2v2
v1 - v1' = v2

Because you have an arithmetic error here. You should get $v_1 + v_1' = v_2$.