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.

The momentum of the system changes at a rate proportional to the net external force: $\vec{F}_{\text {net }}=\frac{d \vec{p}}{d t}$

A large impulse results in a large change in the object's momentum.

A large area under the force-time graph indicates a large impulse delivered to the object.

Applying a constant net force over time results in a uniform change in momentum.

The object experiences a change in momentum and exerts an equal and opposite impulse on the wall.

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.