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4.2 Change in Momentum and Impulse

Syllabus
2024
Topic
4.2
Level
—

Learning objectives

4.2A—Describe the impulse delivered to an object or systemDescribe the impulse delivered to an object or system.• The rate of change of momentum is equal to the net external force exerted on an object or system. Relevant equation:• Impulse is defined as the product of the average force exerted on a system and the time interval during which that force is exerted on the system. Relevant equation:• Impulse is a vector quantity and has the same direction as the net force exerted on the system.• The impulse delivered to a system by a net external force is equal to the area under the curve of a graph of the net external force exerted on the system as a function of time.• The net external force exerted on a system is equal to the slope of a graph of the momentum of the system as a function of time.4.2B—Describe the relationship between the impulse exerted on an object or a system and the change in momentum of the…Describe the relationship between the impulse exerted on an object or a system and the change in momentum of the object or system.• Change in momentum is the difference between a system’s final momentum and its initial momentum. Relevant equation:• The impulse–momentum theorem relates the impulse exerted on a system and the system’s change in momentum. Relevant equation:• Newton’s second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass. Relevant equation BOUNDARY STATEMENT AP Physics 1 does not require students to quantitatively analyze systems in which the mass of the system changes with respect to time. AP Physics 1: Algebra-Based Course and Exam Description Linear Momentum UNIT 4 TOPIC 4.3 Conservation of Linear Momentum | AP Physics 1: Algebra-Based Course and Exam Description

Impulse accumulates force over time

Connect force to momentum rate

A net external force changes an object's or system's momentum. The stronger the force or the longer it acts, the larger the momentum effect.

F⃗net=Δp⃗ΔtJ⃗=F⃗avgΔt\begin{gathered}\vec F_{\text{net}}=\frac{\Delta\vec p}{\Delta t}\\ \vec J=\vec F_{\text{avg}}\Delta t\end{gathered}

Calculate a signed impulse

Signed pulse example: choose right as positive. A net average force of −40 N-40\,\text{N} acts for 0.25 s0.25\,\text{s}.

J=FavgΔt=(−40 N)(0.25 s)=−10 N sJ=F_{\text{avg}}\Delta t=(-40\,\text{N})(0.25\,\text{s})=-10\,\text{N}\,\text{s}.

The impulse is 10 N s10\,\text{N}\,\text{s} leftward, the same direction as the net force.

Map area and slope

Graph What to read Physical meaning
Net external force FnetF_{\text{net}} versus time tt Signed area under the curve Impulse delivered during the interval
System momentum pp versus time tt Slope Δp/Δt\Delta p/\Delta t Net external force during the interval

Keep direction and graph roles

Impulse is a vector, not merely force magnitude times time. Area below the time axis gives impulse in the negative direction. On a momentum–time graph it is the slope, not the area, that represents net external force.

Impulse equals change in momentum

Subtract final and initial vectors

Change in momentum is a vector subtraction: final momentum minus initial momentum. Choose a positive direction before substituting one-dimensional velocities.

Δp⃗=p⃗f−p⃗iJ⃗=F⃗avgΔt=Δp⃗\begin{gathered}\Delta\vec p=\vec p_f-\vec p_i\\ \vec J=\vec F_{\text{avg}}\Delta t=\Delta\vec p\end{gathered}

Calculate a reversal

Direction-reversal example: choose right as positive. A 0.50 kg0.50\,\text{kg} object changes velocity from +4.0 m/s+4.0\,\text{m/s} to −2.0 m/s-2.0\,\text{m/s}.

Δp=m(vf−vi)=(0.50 kg)(−2.0−4.0) m/s=−3.0 kg m/s\Delta p=m(v_f-v_i)=(0.50\,\text{kg})(-2.0-4.0)\,\text{m/s}=-3.0\,\text{kg}\,\text{m/s}.

Therefore J=−3.0 N sJ=-3.0\,\text{N}\,\text{s}: the impulse is leftward.

Recover Newton's second law

For a system of constant mass,

F⃗net=Δp⃗Δt=mΔv⃗Δt=ma⃗.\vec F_{\text{net}}=\frac{\Delta\vec p}{\Delta t}=m\frac{\Delta\vec v}{\Delta t}=m\vec a.

Newton's second law therefore follows directly from the impulse–momentum theorem when mass does not change.

Keep the mass condition

When velocity reverses, subtract signed vectors—not speed magnitudes. AP Physics 1 does not require quantitative analysis of systems whose mass changes with time, so the step Δp⃗=mΔv⃗\Delta\vec p=m\Delta\vec v is used here only for constant mass.