# Is there conservation of momentum if there's conservation of energy? [closed]

The equation for conservation of momentum: $$m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{u}_1 + m_2\vec{u}_2$$

The equation for conservation of energy: $$\frac 12m_1v_1^2 + \frac 12m_2v^2_2 = \frac 12m_1u_1^2 + \frac 12m_2u^2_2$$

Then if it's an elastic collision on one dimension (conservation of energy and also conservation of momentum): $$v_1 - v_2 = -(u_1 - u_2)$$

My question is:

If the case is that there is conservation of energy, will there be conservation of momentum?

In both cases, can you explain why?

Also, let's say they give you a question where there is a collision.

How can you prove there is conservation of energy?

Sometimes the questions tell you that a collision was elastic. But if not, how do you prove it?

• Hint: Think about a moving particle reversing direction. – WillO Mar 12 '16 at 18:09
• @WillO You are talking about the prove right? So I guess it will be proving that $\vec{v} = -\vec{u}$ – Pichi Wuana Mar 12 '16 at 18:21
• Suggestion: You might want to compare the harmonic oscillator with the title question (v1). – Qmechanic Mar 12 '16 at 19:05
• Momentum is always conserved. And for a closed system with no outside forces, the momentum will be constant for a given reference frame. – Bill N Mar 13 '16 at 1:27

• Not all the collisions are with momentum conserved. For example, in a free fall. My condition for conservation of momentum is that all external forces are equal to $0$ or that the net force of external forces is equal to $0$. ($W$ is an external force). – Pichi Wuana Mar 12 '16 at 19:57