Momentum

Momentum is mass in motion, and is represented by the variable $p$ with the unit $\frac{\text{kg} \times \text{m}}{\text{s}}$¹⁾. Each property of momentum is transcribed from each one of Newton's Laws of Motion.

Recall that the first law is the law of inertia and is in effect when $F_{net} = 0$. In these situations where momentum is constant, it is equal to mass times velocity.

• $p = mv$ where $p$ is momentum, $m$ is mass²⁾, and $v$ is velocity³⁾.

The second law of motion deals with changing velocities, or where $F_{net} \ne 0$, and as such momentum here is represented as a change. Change in momentum is also equal to force times time elapsed, which is derived from the force equation $F = ma$.

• $\Delta p = m\Delta v$ where $\Delta p$ is the change in momentum, $m$ is still mass, and $\Delta v$ is the change in velocity.
• $\Delta p = Ft$ where $\Delta p$ is change in momentum, $F$ is force⁴⁾, and $t$ is time⁵⁾.

These two equations can be substituted together to solve for mass, velocity, force, or time given the other three (i.e. $m\Delta v = Ft$).

The third law states that for every force there is an equal and opposite force. This applies to momentum aswell, however instead of forces it is momentum impulses that hold an equal and opposite reaction.

• $\Delta p = -\Delta p$, in other words, whatever momentum one object loses the other one gains.

The collision formula can be used to calculate the masses or velocities of two or more objects after they collide. Think of it as an extension to the laws of motion. The formula states that the sum of the momentum of the objects before and after the collisions are equal. Of course, this assumes that the change in momentum is zero.

• $\Delta p_f = \Delta p_i$ when $\Delta p = 0$

To calculate the initial and final momentums, substitute each $\Delta p$ with $p = mv$.

• $\Delta p_f = m_av_a + m_bv_b + ...$, where the masses and velocities are the final values.
• $\Delta p_i = m_av_a + m_bv_b + ...$, where the masses and velocities are the initial values.

Then, as $\Delta p_f = \Delta p_i$ you can substitute them with the above substitutions to get a solvable equation for any missing values.

• $m_av_a + m_bv_b + ... = m_av_a + m_bv_b + ...$, one side has the initial values, and the other has the final.

A composite system is where the combined speed is given for a set of objects after collision. The momentum here would be equal to the combined masses multiplied by the combined velocities. Instead using the above $\Delta p_f$ equation, use the below one to make your collision equation. Since the combined speed is what is typically given, velocity is represented with just one $v$, though it could also be a sum of velocities in parenthesis.

• $\Delta p_f = (m_a + m_b + ...)v$