Momentum

The Law of Conservation of Momentum states that in a closed system (where no external forces like friction act), the total momentum before an event is equal to the total momentum after the event.

An 'event' typically refers to a collision or an explosion. Even if individual objects change their velocities, the vector sum of all momenta remains constant.

This principle allows us to predict the resulting velocities of objects after they interact by setting the sum of $m \times v$ for all parts before the event equal to the sum after the event.

Step 1: Define Direction: Assign a positive direction (usually right) and a negative direction (usually left). Ensure all velocity values follow this convention.

Step 2: Calculate Initial Momentum: Sum the momenta of all objects before the collision ($p_{total, initial} = m_1u_1 + m_2u_2$).

Step 3: Express Final Momentum: Sum the momenta after the collision ($p_{total, final} = m_1v_1 + m_2v_2$). If objects stick together, use their combined mass $(m_1 + m_2) \times V_{final}$.

Step 4: Solve for Unknowns: Equate the initial and final totals to find the missing velocity or mass.

Newton's Second Law can be expressed in terms of momentum: Force is equal to the rate of change of momentum.

This relationship is defined by the formula $F = \frac{\Delta p}{\Delta t}$, where $\Delta p$ is the change in momentum (final - initial) and $\Delta t$ is the time taken for the change.

This principle explains why a larger force is required to stop an object quickly compared to stopping it slowly over a longer duration.

Check Your Signs: The most common error is forgetting that velocity is a vector. If two objects move toward each other, one MUST have a negative velocity in your equation.

Unit Consistency: Always ensure mass is in $kg$ and velocity is in $m/s$ before calculating. If given grams or $km/h$, convert them first.

Sanity Check: After a collision where objects stick together, the final velocity should generally be between the two initial velocities and the system should move in the direction of the object that had the higher initial momentum.

Safety Questions: When asked about safety features (airbags, seatbelts), always structure your answer around: 1. Increasing the time of impact ($\Delta t$), 2. Which decreases the rate of change of momentum, 3. Resulting in a smaller force ($F$) on the person.