# Which should I conserve? Energy, or Momentum?

Suppose we have two bodies. One with mass $m_{1}$ and another with mass $m_{2}$, such that $m_{1}\neq m_{2}$

The first body is moving along a straight line with some speed $v$ and is destined to collide with the second body, which is perfectly stationary somewhere along that line.

The initial energy and momentum of the first body are

$E_{1}=\dfrac{m_{1} v^2}{2}$

$p_{1}=m_{1} v$

And the initial energy and momentum of the second body are

$E_{2}=0$

$p_{2}=0$

When the two bodies finally collide, the second body takes off along the same straight line at some speed $u$ and the first body remains stationary.

Conservation of energy would then require that

$\dfrac{m_{1} v^2}{2}=\dfrac{m_{2} u^2}{2}$

$v^2=\dfrac{m_{2}}{m_{1} } u^2$

On the other hand, conservation of momentum requires that

$m_{1} v=m_{2} u$

$v=\dfrac{m_{2}}{m_{1}} u$

So, if we try to conserve both energy and momentum, we get

$\left(\dfrac{m_{2}}{m_{1}}\right)^2=\dfrac{m_{2}}{m_{1}}$

This would force us to conclude that

$\dfrac{m_{2}}{m_{1}}=0$

or

$\dfrac{m_{2}}{m_{1}}=1$

So, the only way that we can conserve both is if one of the two bodies is mass-less or the two bodies have the same mass. But this violates the condition of two arbitrary masses.

So, we can only conserve either energy, or momentum. But which one? and why?

Or is there something wrong with the claim that the first body $m_{1}$ can be stationary after the collision? If so, what exactly is wrong with this claim?

• "And the first body remains stationary." So, you're imposing this extra condition which may not be consistent with the conservation laws. – Count Iblis Oct 29 '16 at 23:31
• If you need to choose, choose momentum. As long as every force has a counterforce in your system, total momentum is invariant. Energy: not so much. – Andras Deak Oct 29 '16 at 23:32
• Yes, I am imposing it. I want to know if there is something wrong with it. And if there is, I'd like to know what it is. – Mahlomola Daniel Cwele Oct 29 '16 at 23:32
• – Count Iblis Oct 29 '16 at 23:34
• @AndrasDeak But why would there be any extra energy in the system. Where does this energy come from. Isn't that a violation? – Mahlomola Daniel Cwele Oct 29 '16 at 23:37

1) $m_1=m_2$
2) some energy is dissipated during the collision (and $m_1<m_2$)