AP Physics 1 Unit 4 Notes: Understanding Momentum and Impulse

Linear momentum

A vector measure of an object’s “quantity of motion” in a straight line, defined by (\vec p=m\vec v).

Momentum equation (definition)

(\vec p=m\vec v), where mass is a scalar and velocity is a vector, so momentum points in the same direction as velocity.

Vector (in the context of momentum)

A quantity with both magnitude and direction; momentum is a vector, so opposite directions can cancel when added.

Scalar (contrast with momentum)

A quantity with magnitude only; kinetic energy is scalar, unlike momentum.

Momentum components

The x- and y-direction parts of momentum (e.g., (px, py)) used to track momentum in two dimensions.

Sign convention in 1D momentum

A method to represent direction using positive/negative signs (e.g., right positive implies leftward momentum is negative).

SI units of momentum

(\mathrm{kg\cdot m/s}), obtained from (m) in kg and (v) in m/s.

Kinetic energy vs momentum (key difference)

Kinetic energy is scalar and depends on (v^2); momentum is vector and depends on (v), so momentum can cancel by direction while energy cannot.

System (in momentum problems)

A chosen collection of objects analyzed together to track total momentum and identify internal vs external forces.

Internal forces

Forces objects within a system exert on each other; they can change individual momenta but tend to cancel in pairs for the system total (Newton’s third law idea).

External forces

Forces exerted on a system by objects outside the system; these can change the system’s total momentum.

Total momentum (of a system)

The vector sum of all objects’ momenta in the chosen system; affected by external forces during an interaction.

Impulse

A vector measure of a force acting over a time interval; for constant/average net force, (\vec J=\vec F_{\text{net,avg}}\Delta t).

Impulse–momentum theorem

(\vec J=\Delta\vec p); impulse equals the change in momentum of an object/system over the interaction time.

Change in momentum

Final minus initial momentum: (\Delta\vec p=\vec pf-\vec pi); direction matters because momentum is a vector.

Average net force

The net force averaged over the interaction time, used in (\vec J=\vec F_{\text{net,avg}}\Delta t).

Momentum form of Newton’s second law (conceptual bridge)

(\vec F_{\text{net}}=\frac{\Delta\vec p}{\Delta t}), emphasizing that net force changes momentum over time.

Units of impulse

(\mathrm{N\cdot s}), which is equivalent to (\mathrm{kg\cdot m/s}) because impulse equals change in momentum.

Impulse as area under an (F)–(t) graph

Impulse over a time interval equals the signed area under the net force vs time curve (rectangles, triangles, trapezoids).

Negative impulse (meaning)

An impulse with negative sign indicates the net force (and momentum change) is in the negative direction of the chosen axis.

Increasing collision time reduces average force

For a fixed (\Delta p), (F_{\text{avg}}=\frac{\Delta p}{\Delta t}), so a larger (\Delta t) results in a smaller average net force magnitude.

Bounce vs stop (momentum change)

Reversing direction (bouncing) produces a larger (|\Delta p|) than simply stopping from the same initial speed (often double if speed is the same in opposite directions).

Common mistake: using speed instead of velocity

Dropping direction causes sign/direction errors because momentum and (\Delta p) depend on velocity (a vector), not speed (a scalar).

Common mistake: mixing momentum and force

Momentum describes motion state ((\vec p)); force describes interaction that changes momentum (via impulse or (\Delta\vec p)).

Common mistake: wrong (\Delta p) order

Writing (\Delta p=pi-pf) instead of the correct (\Delta p=pf-pi), leading to sign errors.