Calculating Final Velocities in a Head-On Collision | Physics Homework Solution

[kraigandrews]

Homework Statement

A mass m with initial velocity v0 collides head on with a mass 2m initially at rest. What is the final velocity of the smaller mass, as a ratio to v0?

What is the final velocity of the larger mass, as a ratio to v0?

Homework Equations

Pi=Pf

KE=.5mv^2

The Attempt at a Solution

for part a I am confused as to what I am doing wrong:

mv_o=mv₁+2mv₂
it wants to find v1 in terms of vo so:

v₂=.5(v_o+v₁)

then plug into KE equation:

v_o²=v₁²+.25(v₁²+v_o²)
and simplify to get

(3/4)v_o²=(5/4)v₁²

then v₁=(3/5)^1/2v_o

 

[ehild]

for part a I am confused as to what I am doing wrong:

mv_o=mv₁+2mv₂
it wants to find v1 in terms of vo so:

v₂=.5(v_o+v₁)

That + should be minus.


ehild

 

[tiny-tim]

hi kraigandrews!

mv_o=mv₁+2mv₂
it wants to find v1 in terms of vo so:

v₂=.5(v_o+v₁)

then plug into KE equation:

v_o²=v₁²+.25(v₁²+v_o²)
and simplify to get

(3/4)v_o²=(5/4)v₁²

then v₁=(3/5)^1/2v_o

hmm …

i] v₂=.5(v_o minus v₁)

ii] you need to put m or 2m in the KE equation

iii] (v_o+v₁)² is not v₁²+v_o²

 

[kraigandrews]

yeah wow that was a dumb mistake not foiling, for the KE part I just canceled the masses:

but now I am left with
(3/4)Vo^2=(5/4)V1^2-(1/2)Vo*V1
and I am not quite sure how to solve this for V1

 

[tiny-tim]

erm … it's a quadratic equation!

(and anyway, v₀ = v₁ is obviously going to be a solution, since that corresponds to the masses not colliding!)