Momentum

• The momentum of an object with the mass $m$ and the velocity $\vec{v}$ is defined as $$\vec{p} = m\vec{v}$$
• The total momentum of all objects in a closed system is always conserved.
• The derivative of $\vec{p}$ with respect to time equals the force acting on an object $$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$

If two objects interact, the absolute values of the forces acting on both objects are equal at any given time due to Newton's third law: $$\vec{F}_{1,2}(t) = -\vec{F}_{2,1}(t)$$ If we are interested in the velocities of the objects, we can replace $\vec{F}$ with $m\vec{a}$ according to Newton's second law and integrate over from a given time $t$ to $t'$: $$m_1\int_{t}^{t'}\vec{a}_1(t)\,\mathrm{d}t = -m_2\int_{t}^{t'}\vec{a}_2(t)\,\mathrm{d}t$$ Integrating both sides leads to: $$m_1(\vec{v}_1' - \vec{v}_1) = -m_2(\vec{v}_2' - \vec{v}_2)$$ This equation can be rearranged according to $$m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}_1' + m_2\vec{v}_2'$$ This leads to the following important definition of the momentum of an object: Inserting this definition into the equation mentioned above results in the following statement. $$\vec{p}_1 + \vec{p}_2 = \vec{p}_1' + \vec{p}_2'$$ It says for instance that the sum of the momenta before the collision is equal to the sum of the momenta after the collision as long as now external forces are applied. For the general case of $N$ interacting bodies, this equation can be extended to the following statement. Therefore, the momentum is similar to the energy of a conserved quantity in a closed system as long as no external force is applied. Since momentum is defined as the product of mass and velocity, the force acting on an object, is equal to the change in momentum with respect to time. The integral of $\vec{F}$ over time $${\Delta p = \int_{t_1}^{t_2} F\,\mathrm{d}t}$$ is called impulse.