Change in Momentum and Impulse - AP Physics 1: Algebra-Based Study Guide

Getting Started

This chapter explores interactions that happen over very short periods, like a tennis racket striking a ball or a foot kicking a soccer ball. We will focus on the system of a single object undergoing a collision. The core question we will answer is: How can we analyze the change in an object's motion when the forces involved are immense but act only for a brief instant?

What You Should Be Able to Do

After studying this section, you should be able to:

• Define impulse as a vector quantity representing the product of an average net force and the time interval over which it acts.

• Describe how an impulse delivered to an object causes a change in that object's momentum.

• Apply the impulse-momentum theorem to solve problems involving collisions and short-duration forces.

• Explain the relationship between Newton's second law and the impulse-momentum theorem.

Key Concepts & Mechanisms

System & Preconditions

To analyze a collision, we first define our system, which is the object or collection of objects we are interested in (e.g., the baseball). Everything outside the system is the environment (e.g., the bat, the air, Earth). The key interaction we study is a force exerted by the environment on the system over a specific time interval.

Our primary idealization is to focus on the net external force (${\vec{F}}_{net}$) that causes the change in motion. During a brief, intense collision, like a bat hitting a ball, the force of the impact is often so much larger than other forces (like gravity or air resistance) that we can temporarily ignore the others. The force during the collision is rarely constant, so we typically work with the average net force (${\vec{F}}_{avg}$).

Key Steps & Relations

1. Newton's Second Law, Revisited: We can express Newton's second law in terms of momentum. The net external force on a system is equal to the rate at which its momentum changes.

   ${\vec{F}}_{net}=\frac{\Delta \vec{p}}{\Delta t}$

   This is the most general form of the second law. It states that a net force is the cause of a change in momentum over time.

2. Defining Impulse: To better analyze the effect of a force acting over a time interval, we define a new quantity called impulse. Impulse ($\vec{J}$) is the product of the average net force exerted on a system and the time interval during which that force acts.

   $\vec{J}={\vec{F}}_{avg}\Delta t$

   Impulse is a vector quantity that has the same direction as the average net force. Its units are Newton-seconds (N·s).

3. The Impulse-Momentum Theorem: By rearranging the momentum form of Newton's second law (${\vec{F}}_{net}=\frac{\Delta \vec{p}}{\Delta t}$), we get ${\vec{F}}_{net}\Delta t=\Delta \vec{p}$. We can now substitute our definition of impulse into this equation to arrive at a powerful new relationship.

   $\vec{J}=\Delta \vec{p}$

   This critical relationship is called the impulse-momentum theorem. It states that the impulse delivered to a system is exactly equal to the change in that system's momentum. An impulse is not something a system has; it is something that is done to a system, resulting in a change in its momentum.

4. Connecting to Kinematics: For a system with constant mass ($m$), the change in momentum is $\Delta \vec{p}=m{\vec{v}}_f-m{\vec{v}}_i=m({\vec{v}}_f-{\vec{v}}_i)=m\Delta \vec{v}$. Substituting this into the impulse-momentum theorem gives:

   ${\vec{F}}_{avg}\Delta t=m\Delta \vec{v}$

   If we divide by $\Delta t$, we get ${\vec{F}}_{avg}=m\frac{\Delta \vec{v}}{\Delta t}$. Since acceleration is $\vec{a}=\frac{\Delta \vec{v}}{\Delta t}$, this returns us to the familiar form ${\vec{F}}_{net}=m\vec{a}$. This shows that the impulse-momentum theorem is a direct and logical consequence of Newton's second law.

Outputs & Effects

The primary effect of an external impulse on a system is a change in the system's momentum. For an object of constant mass, this means its velocity changes. A key insight is that the same change in momentum can be achieved in different ways: a large average force acting for a short time (like a hammer hitting a nail) can produce the same impulse as a small average force acting for a long time (like the gentle, constant push of a rocket engine in space).

Regulation & Limits

• Average vs. Instantaneous Force: In most real-world collisions, the force varies dramatically over the time interval. The ${\vec{F}}_{avg}$ in the impulse equation is the average value of this force. The impulse-momentum theorem allows us to calculate this average force without needing to know the details of how the force changed moment-to-moment.

• Vector Nature: Impulse and momentum are vectors. When solving problems, you must account for direction. For one-dimensional motion, this means assigning positive and negative signs to velocities and forces. A common case is an object bouncing, where its initial and final velocities have opposite signs.

• Net External Force: The theorem applies to the net external force. If multiple forces act on the system, ${\vec{F}}_{avg}$ is the vector sum of all average external forces.

Key Models & Diagrams

The relationship between force, time, momentum, and impulse can be summarized in the following way:

Key Components & Evidence

• Momentum ($\vec{p}$): The product of an object's mass and velocity ($\vec{p}=m\vec{v}$). It is a vector measure of an object's motion. Units: kg·m/s.

• Impulse ($\vec{J}$): The product of the average net force on an object and the time interval over which it acts. It is a vector measure of the "punch" delivered by a force. Units: N·s.

• Net Force (${\vec{F}}_{net}$): The vector sum of all forces acting on an object. A net force is required to change an object's momentum. Units: Newtons (N).

• Time Interval ($\Delta t$): The duration over which the net force is applied. Units: seconds (s).

• Change in Momentum ($\Delta \vec{p}$): The final momentum minus the initial momentum. This is the direct result of a net impulse. Units: kg·m/s.

• Impulse-Momentum Theorem ($\vec{J}=\Delta \vec{p}$): The core law connecting these quantities. It is a restatement of Newton's second law.

• Lab Observation: In the lab, you can use a force sensor and a motion detector to measure the force on a cart during a collision and its change in velocity. The area under the force-time graph (the impulse) will be equal to the cart's mass times its change in velocity (the change in momentum).

Skill Snapshots

• Causation:

  1. A net external force applied over a time interval causes an impulse.

  2. An impulse delivered to an object causes its momentum to change.

  3. For a constant-mass object, a change in momentum causes a change in velocity.

• Comparison:

  1. Impulse vs. Force: Impulse is the effect of a force over time, while force is the instantaneous push or pull. A small force for a long time can deliver a larger impulse than a large force for a short time.

  2. Impulse vs. Work: Impulse ($\vec{F}\Delta t$) causes a change in momentum ($\Delta \vec{p}$), while Work ($F\Delta x$) causes a change in kinetic energy ($\Delta K$). Both involve force, but impulse is concerned with the time of application, while work is concerned with the distance of application.

  3. Momentum vs. Kinetic Energy: Momentum ($\vec{p}=m\vec{v}$) is a vector that describes the quantity of motion, while kinetic energy ($K=\frac{1}{2}m{v}^2$) is a scalar that describes the energy of motion. An object can have negative momentum but never negative kinetic energy.

• Change Over Time:

  • Baseline: An object has an initial momentum, ${\vec{p}}_i=m{\vec{v}}_i$.

  • Change 1: An average net force, ${\vec{F}}_{avg}$, is applied for a duration $\Delta t$, delivering an impulse $\vec{J}$.

  • Change 2: The object's momentum changes by an amount $\Delta \vec{p}=\vec{J}$, resulting in a new final momentum ${\vec{p}}_f={\vec{p}}_i+\vec{J}$.

  • Continuity: In the absence of a net external force (i.e., if $\vec{J}=0$), the object's momentum does not change ($\Delta \vec{p}=0$).

Common Misconceptions & Clarifications

1. Misconception: Impulse is the same as force.

   • Clarification: Impulse is not just force; it is the product of the average force and the time interval over which it acts ($\vec{J}={\vec{F}}_{avg}\Delta t$). To reduce the force experienced during a collision with a given change in momentum, you must increase the time of the collision (e.g., airbags, bending your knees when landing).

2. Misconception: The change in momentum is just the final momentum ($m{\vec{v}}_f$).

   • Clarification: Change always means final minus initial. The change in momentum is $\Delta \vec{p}={\vec{p}}_f-{\vec{p}}_i$. If an object bounces, its initial and final velocities have opposite signs, leading to a larger change in momentum than if it had just stopped.

3. Misconception: Momentum and impulse are scalars.

   • Clarification: Both are vector quantities. Direction is crucial. In one-dimensional problems, you must use positive and negative signs to indicate direction. A positive impulse can cause a negative momentum to become less negative or even positive.

4. Misconception: An object "has" impulse.

   • Clarification: An object has momentum. Impulse is not a property of an object; it is a process done to an object by an external force. It is the transfer of momentum to or from the object.

One-Paragraph Summary

The impulse-momentum theorem provides a powerful framework for analyzing collisions and other interactions involving forces that act over time. Impulse, defined as the product of the average net external force and the time interval of its application, is a vector quantity that quantifies the overall effect of that force. The theorem states that this impulse is precisely equal to the change in the system's momentum. This relationship, which is a direct consequence of Newton's second law, is especially useful in situations where forces are complex and short-lived, as it allows us to relate the easily measured change in velocity of an object to the average force that caused it. By understanding impulse, we can explain why extending the time of an impact, such as through an airbag, is effective at reducing the force experienced.