The impulse is equal to the area under the force-time graph between the initial and final times.

The net external force is equal to the slope of the momentum-time graph at a given point in time.

1. Determine the initial momentum ($\vec{p}_0$). 2. Determine the final momentum ($\vec{p}$). 3. Subtract the initial momentum from the final momentum: $\Delta \vec{p}=\vec{p}-\vec{p}_{0}$

1. Identify the net force as a function of time, $\vec{F}_{\text {net }}(t)$. 2. Determine the time interval $[t_1, t_2]$. 3. Integrate the force function over the time interval: $\vec{J}=\int_{t_{1}}^{t_{2}} \vec{F}_{\text {net }}(t) d t$.

1. Identify the impulse acting on the object. 2. Identify the initial and final momentum of the object. 3. Set the impulse equal to the change in momentum: $\vec{J} = \Delta \vec{p}$. 4. Solve for the unknown quantity.

Impulse: Change in momentum due to force acting over time, measured in N⋅s. | Momentum: Mass in motion, measured in kg⋅m/s.

Large force, short time: Results in a large impulse if the product of force and time is significant. | Small force, long time: Can also result in a large impulse if the product of force and time is significant.

Force-time graph: Area under the curve represents impulse. | Momentum-time graph: Slope represents net external force.

Positive change in momentum: Indicates an increase in momentum (object speeds up in the positive direction). | Negative change in momentum: Indicates a decrease in momentum (object slows down or changes direction).

Newton's Second Law: F = ma, applies when mass is constant. | Impulse-Momentum Theorem: J = Δp, more general and applies even when mass is not constant.

Impulse: The change in momentum of an object ($\vec{J} = \Delta \vec{p}$), caused by a force acting over time. | Momentum: The product of an object's mass and velocity ($\vec{p} = m\vec{v}$), representing its inertia in motion.

Force-time graph: Area under the curve represents the impulse. | Momentum-time graph: Slope of the curve represents the net force.

Constant mass: $\vec{F}_{\text {net }} = m\vec{a}$ applies, where mass is constant. | Variable mass: $\vec{F}_{\mathrm{net}}=\frac{d \vec{p}}{d t}=\frac{d m}{d t} \vec{v}$ applies, considering the rate of mass change.