# Linear Momentum

The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as

Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum $\mathbf{p}$ is a vector having the same direction as the velocity $\mathbf{\text{v}}$. The SI unit for momentum is $\text{kg}\xb7\text{m/s}$.

(a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.

**Strategy**

No information is given regarding direction, and so we can calculate only the magnitude of the momentum, $p$. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes

when only magnitudes are considered.

**Solution for (a)**

To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.

**Solution for (b)**

To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.

The ratio of the player’s momentum to that of the ball is

**Discussion**

Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.

# Momentum and Newton’s Second Law

The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is

where ${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta}\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta}t$ is the change in time.

This statement of Newton’s second law of motion includes the more familiar ${\mathbf{F}}_{\text{net}}\text{=}m\mathbf{a}$ as a special case. We can derive this form as follows. First, note that the change in momentum $\mathrm{\Delta}\mathbf{p}$ is given by

If the mass of the system is constant, then

So that for constant mass, Newton’s second law of motion becomes

Because $\frac{\mathrm{\Delta}\mathbf{v}}{\mathrm{\Delta}t}=\mathbf{a}$, we get the familiar equation

*when the mass of the system is constant*.

Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail**;** however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.

During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?

**Strategy**

This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as

As noted above, when mass is constant, the change in momentum is given by

In this example, the velocity just after impact and the change in time are given; thus, once $\mathrm{\Delta}p$ is calculated, ${F}_{\text{net}}=\frac{\mathrm{\Delta}p}{\mathrm{\Delta}t}$ can be used to find the force.

**Solution**

To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.

Now the magnitude of the net external force can determined by using ${F}_{\text{net}}=\frac{\mathrm{\Delta}p}{\mathrm{\Delta}t}$:

where we have retained only two significant figures in the final step.

**Discussion**

This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using ${F}_{\text{net}}\phantom{\rule{0.15em}{0ex}}\text{=}\phantom{\rule{0.15em}{0ex}}\text{ma}$, but one additional step would be required compared with the strategy used in this example.

# Section Summary

- Linear momentum (
for brevity) is defined as the product of a system’s mass multiplied by its velocity.*momentum* - In symbols, linear momentum $\mathbf{p}$ is defined to be
$\mathbf{p}=m\mathbf{v},$where
*$m$*is the mass of the system and $\mathbf{v}$ is its velocity. - The SI unit for momentum is $\text{kg}\xb7\text{m/s}$.
- Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
- In symbols, Newton’s second law of motion is defined to be
${\mathbf{F}}_{\text{net}}=\frac{\mathrm{\Delta}\mathbf{p}}{\mathrm{\Delta}t}\text{,}$${\mathbf{F}}_{\text{net}}$ is the net external force, $\mathrm{\Delta}\mathbf{p}$ is the change in momentum, and $\mathrm{\Delta}t$ is the change time.

# Conceptual Questions

An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?

An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

**Professional Application**

Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.

How can a small force impart the same momentum to an object as a large force?

# Problems & Exercises

(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of $7\text{.}\text{50 m/s}$. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of $\text{600 m/s}$. (c) What is the momentum of the 90.0-kg hunter running at $7\text{.}\text{40 m/s}$ after missing the elephant?

(a) $\text{1.50}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$

(b) 625 to 1

(c) $6\text{.}\text{66}\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$

(a) What is the mass of a large ship that has a momentum of $1\text{.}\text{60}\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{kg}\xb7\text{m/s}$, when the ship is moving at a speed of $\text{48.0 km/h?}$ (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of $\text{1200 m/s}$.

(a) At what speed would a $2\text{.}\text{00}\times {\text{10}}^{4}\text{-kg}$ airplane have to fly to have a momentum of $1\text{.}\text{60}\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{kg}\xb7\text{m/s}$ (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of $\text{60.0 m/s}$? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.

(a) $8\text{.}\text{00}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(b) $1\text{.}\text{20}\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg}\xb7\text{m/s}$

(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be $-0\text{.}\text{0100 m/s}$, which is probably not noticeable.

(a) What is the momentum of a garbage truck that is $1\text{.}\text{20}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and is moving at $10\text{.}\text{0 m/s}$? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck?

A runaway train car that has a mass of 15,000 kg travels at a speed of $5\text{.4 m/s}$ down a track. Compute the time required for a force of 1500 N to bring the car to rest.

54 s

The mass of Earth is $5\text{.}\text{972}\times {10}^{\text{24}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and its orbital radius is an average of $1\text{.}\text{496}\times {10}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{m}$. Calculate its linear momentum.

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation