⚾️Honors Physics Unit 8 Review

Linear momentum ( $p$ ) measures the quantity of motion an object has, combining how massive it is with how fast it's moving. The formula is straightforward:

$p = mv$

where $m$ is mass and $v$ is velocity. Momentum is a vector quantity, so it has both magnitude and direction. A 2 kg ball moving east at 5 m/s has a momentum of 10 kg·m/s east.

A heavier object moving at the same speed as a lighter one carries more momentum. Likewise, a faster object carries more momentum than a slower one of the same mass. This is why a slow-moving freight train is harder to stop than a fast-moving tennis ball.

Impulse

Impulse ( $J$ ) describes the effect of a force acting over a period of time. It's calculated as:

$J = F_{net} \Delta t$

Impulse is also a vector. If you push on something with 45 N for 2 seconds, the impulse is 90 N·s in the direction of the force.

The key relationship is that impulse equals the change in momentum of an object:

$J = \Delta p = p_f - p_i$

Think of impulse as the "push" that reshapes an object's momentum. A baseball bat contacting a ball for a brief instant delivers an impulse that dramatically changes the ball's momentum from roughly zero to very large in the direction of the hit.

Concept of linear momentum, Chapter 9: Linear Momentum – Introductory Physics Resources

Impulse-Momentum Theorem Applications

The impulse-momentum theorem ties everything together:

$J = \Delta p = m(v_f - v_i)$

This says the impulse on an object equals its mass times its change in velocity. To solve problems with this theorem:

Identify initial and final velocities ( $v_i$ and $v_f$ ). For example, a 1,500 kg car traveling at 20 m/s comes to a stop, so $v_i = 20$ m/s and $v_f = 0$ m/s.
Determine the mass ( $m$ ) of the object: 1,500 kg.
Calculate the change in momentum: $\Delta p = m(v_f - v_i) = 1{,}500(0 - 20) = -30{,}000$ kg·m/s. The negative sign means momentum decreased in the original direction of motion.
If force or time is given, use $J = F_{net} \Delta t$ to find the unknown. If the brakes apply force over 5 seconds: $F_{net} = \frac{\Delta p}{\Delta t} = \frac{-30{,}000}{5} = -6{,}000$ N.

This theorem is your go-to tool for analyzing collisions, explosions, rocket launches, and any scenario where forces act over time.

One useful insight: the same change in momentum can result from a large force over a short time or a small force over a long time. This is exactly why airbags work. They extend the time of impact during a crash, reducing the peak force on your body while producing the same impulse.

Newton's Second Law and Momentum Change

Newton's second law is usually written as $F_{net} = ma$ , but it can be rewritten in a more general momentum form:

$F_{net} = \frac{\Delta p}{\Delta t}$

This says the net force on an object equals the rate of change of its momentum. The two forms are equivalent for constant mass (since $\frac{\Delta(mv)}{\Delta t} = m\frac{\Delta v}{\Delta t} = ma$ ), but the momentum version is more powerful because it also handles situations where mass changes, like a rocket burning fuel.

Several important consequences follow from this:

No net force means no change in momentum. If $F_{net} = 0$ , then $\Delta p = 0$ , so momentum stays constant. A spacecraft drifting in deep space with no forces acting on it maintains the same momentum indefinitely.
Greater net force means a faster change in momentum. A tennis player swinging harder (more force) changes the ball's momentum more quickly, sending it off at a higher speed.
The direction of the net force determines the direction of the momentum change. A braking force opposite to motion reduces momentum in the forward direction.

This momentum formulation of Newton's second law is the foundation for everything you'll study in this unit, from conservation of momentum to collision analysis.

⚾️Honors Physics Unit 8 Review