Linear Momentum Defined

Momentum is the measure of an object's mass in motion. Any moving body possesses linear momentum $p$ , defined as the product of its inertial mass $m$ and instantaneous velocity $v$ :

p = mv

Because velocity is a vector quantity, momentum is also a vector pointing in the direction of motion. In standard SI units, momentum is expressed in kilogram meters per second ( $\text{kg}\cdot\text{m/s}$ ) or Newton-seconds ( $\text{N}\cdot\text{s}$ ).

Impulse: Changing Momentum

To change an object's momentum, an external net force must act over an interval of time. Applying Newton's Second Law ( $\Sigma F = ma = m\frac{\Delta v}{\Delta t}$ ), we derive the Impulse-Momentum Theorem:

J = F_{\text{avg}}\Delta t = \Delta p = m v_f - m v_i

Here, Impulse (J) represents the cumulative effect of a force applied over a specific time duration (Δ $\Delta t$ ).

The Inverse Relationship of Force and Time

When an object moving at a given velocity comes to a complete rest, its total change in momentum (Δ $\Delta p$ ) is completely fixed by its initial mass and velocity. Consequently, the total impulse ( $J$ ) required to stop the object is constant:

\Delta p = \text{Constant} \implies F_{\text{avg}} = \frac{\Delta p}{\Delta t}

Short Impact Duration (small $\downarrow \Delta t$ ): Results in a massive average impact force (Big $\uparrow F_{\text{avg}}$ ), producing structural crushing, shattered cargo, or catastrophic failure.
Extended Impact Duration (Big $\uparrow \Delta t$ ): Spreads the momentum transfer over a longer duration, dramatically decreasing the peak impact force (small $\downarrow F_{\text{avg}}$ ).

Packaging materials, vehicle crumple zones, air bags, and freight shock absorbers do not reduce the momentum change; they manipulate the time variable (Δ $\Delta t$ ) to suppress the peak destructive force.

Car A has no crumple zone and so the collision happens in a small time with a large force, while car B has a crumple zone and so the collision takes more time with a smaller force.

A graph showing the force versus time curve for two cars. The curve for car A is is narrow covering a small amount of time on the x axis, but rising to a large peak force on the y axis. The curve for car B is broad covering a lot of time on the x axis and rising to a smaller peak force on the y axis.

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