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This page defines the linear momentum of particles and systems.
<big><b>Edwyn Torres — Spring 2026</b></big>
 
=Linear Momentum=
 
==Introduction==
 
Linear momentum is a vector quantity that describes the motion of an object. Momentum depends on two things: the object's mass and the object's velocity. An object with more mass or more velocity has more momentum.
 
Momentum is useful because it helps describe how difficult it is to stop or change the motion of an object. For example, a large truck and a small bicycle can move at the same speed, but the truck has much more momentum because it has much more mass.


==The Main Idea==
==The Main Idea==


Linear momentum is a vector quantity describing an object's motion. It is defined as the product of an object's [[Mass]] (<math>m</math>) and [[Velocity]] (<math>\vec{v}</math>). Note that mass is a scalar while velocity is a vector, so an object's linear momentum is always in the same direction as its velocity. Linear momentum is a vector quantity. Linear momentum is represented by the letter <math>\vec{p}</math> and is often referred to as simply "momentum." The most commonly used metric unit for momentum is the kilogram*meter/second. The plural of momentum is momenta or momentums.
The linear momentum of an object is defined as:
 
<math>\vec{p} = m\vec{v}</math>
 
where:
 
* <math>\vec{p}</math> = linear momentum
* <math>m</math> = mass
* <math>\vec{v}</math> = velocity
 
Since velocity is a vector, momentum is also a vector. This means momentum has both magnitude and direction. The direction of an object's momentum is the same as the direction of its velocity.
 
==Key Equations==
 
For one object:
 
<math>\vec{p} = m\vec{v}</math>
 
For a system of multiple objects:
 
<math>\vec{p}_{total} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + ...</math>
 
For momentum change:
 
<math>\Delta \vec{p} = \vec{p}_f - \vec{p}_i</math>
 
Impulse-momentum relationship:
 
<math>\Delta \vec{p} = \vec{F}_{net}\Delta t</math>
 
or:
 
<math>\vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t</math>
 
==Units==
 
The SI unit for momentum is:
 
<math>kg \cdot m/s</math>
 
Momentum should not be measured in newtons. A newton is a unit of force:
 
<math>1N = 1kg \cdot m/s^2</math>
 
Impulse has units of:
 
<math>N \cdot s</math>
 
Since:
 
<math>N \cdot s = kg \cdot m/s</math>
 
impulse and momentum have equivalent units.
 
==Conceptual Explanation==
 
Momentum describes how much motion an object has. If two objects have the same velocity, the object with more mass has more momentum. If two objects have the same mass, the object with greater velocity has more momentum.
 
For example, a bowling ball rolling at 3 m/s has more momentum than a tennis ball rolling at 3 m/s because the bowling ball has more mass. A fast-moving baseball can also have a large amount of momentum because its velocity is high, even though its mass is small.
 
Momentum is especially important in collisions. During a collision, objects can exert large forces on each other over a short time interval. Even if the forces are complicated, momentum can still be used to analyze the motion before and after the collision.
 
==Single Particle Momentum==
 
For a single particle, momentum is calculated using:
 
<math>\vec{p} = m\vec{v}</math>
 
Example:
 
If a 4 kg object moves with velocity <math>\langle 2, 0, 0 \rangle m/s</math>, then its momentum is:
 
<math>\vec{p} = 4\langle 2,0,0 \rangle</math>
 
<math>\vec{p} = \langle 8,0,0 \rangle kg \cdot m/s</math>
 
This means the object has momentum in the positive x-direction.
 
==Multiple Particle Systems==
 
For a system with more than one object, the total momentum is found by adding the momentum vectors of each object.
 
<math>\vec{p}_{total} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + ...</math>
 
Because momentum is a vector, directions must be included. Momentum in opposite directions can cancel.
 
For example, if one object has momentum:
 
<math>\vec{p}_1 = \langle 10,0,0 \rangle kg \cdot m/s</math>
 
and another object has momentum:
 
<math>\vec{p}_2 = \langle -4,0,0 \rangle kg \cdot m/s</math>
 
then the total momentum is:
 
<math>\vec{p}_{total} = \langle 10,0,0 \rangle + \langle -4,0,0 \rangle</math>
 
<math>\vec{p}_{total} = \langle 6,0,0 \rangle kg \cdot m/s</math>
 
The system still has momentum in the positive x-direction.
 
==Momentum and Impulse==
 
Momentum changes when a net force acts on an object for a period of time. This change in momentum is called impulse.
 
<math>\Delta \vec{p} = \vec{F}_{net}\Delta t</math>
 
A larger force creates a larger change in momentum. A longer time interval also creates a larger change in momentum.
 
This explains why airbags help protect people during car crashes. In a crash, a person's momentum must decrease to zero. An airbag increases the time over which the person slows down. Since the stopping time is longer, the average force on the person is smaller.
 
==Momentum and Newton's Laws==
 
Momentum is closely connected to Newton's laws.
 
Newton's First Law says that an object will continue moving at constant velocity unless acted on by a net external force. In terms of momentum, this means an object's momentum remains constant if the net force is zero.
 
Newton's Second Law can be written using momentum:
 
<math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>
 
This means the net force on an object is equal to the rate of change of its momentum.
 
==Momentum in Rocket Propulsion==
 
Rockets move because of conservation of momentum. A rocket pushes exhaust gases backward at high speed. Because the gases gain momentum backward, the rocket gains momentum forward.
 
This works even in space because the rocket does not need to push against air. Instead, the rocket changes its momentum by ejecting mass in the opposite direction.
 
==Common Misconceptions==
 
===Misconception 1: Momentum and velocity are the same thing===
 
Momentum and velocity are related, but they are not the same. Momentum depends on both mass and velocity. A heavy object moving slowly can have the same momentum as a light object moving quickly.
 
===Misconception 2: Momentum is measured in newtons===
 
Momentum is not measured in newtons. Momentum is measured in:
 
<math>kg \cdot m/s</math>
 
Newtons are used for force.
 
===Misconception 3: Direction does not matter===
 
Momentum is a vector, so direction always matters. A positive momentum and a negative momentum can partially or completely cancel when finding total momentum.


===A Mathematical Model===
===Misconception 4: Momentum only matters in collisions===


====Single Particles====
Momentum is very important in collisions, but it also matters in rocket motion, sports, walking, running, explosions, and systems of particles.
The momentum of a particle is defined as follows:
 
==Worked Example 1: Basic Momentum==
 
A 70 kg person is moving with velocity <math>\langle 1,2,3 \rangle m/s</math>. What is the person's momentum?
 
Use:


<math>\vec{p} = m\vec{v}</math>
<math>\vec{p} = m\vec{v}</math>


where <math>\vec{p}</math> is the particle's linear momentum, <math>m</math> is the particle's mass, and <math>\vec{v}</math> is the particle's velocity. This formula accurately describes the momentum of particles at everyday speeds, but for particles traveling near the speed of light, the formula for [[Relativistic Momentum]] must be used for concepts such as the momentum principle to remain true.
Substitute:


====Multiple Particles====
<math>\vec{p} = 70\langle 1,2,3 \rangle</math>


The total momentum of a system of particles is defined as the vector sum of the momenta of the particles that comprise the system:
Multiply each component by 70:


<math>\vec{p}_{system} = \sum_i \vec{p}_i</math>
<math>\vec{p} = \langle 70,140,210 \rangle kg \cdot m/s</math>


Although the proof does not appear on this page, it can be shown that the total momentum of a system of particles is equal to the total mass of the system times the velocity of its [[Center of Mass]]:
Final answer:


<math>\vec{p}_{system} = M_{tot}\vec{v}_{COM}</math>
<math>\boxed{\vec{p} = \langle 70,140,210 \rangle kg \cdot m/s}</math>


This formula makes it significantly easier to calculate the momentum of certain objects. For example, consider a disk rolling along the ground. The disk is comprised of an infinite number of infinitely small "mass elements," all of which have different momenta; the ones at the bottom of the disk are hardly moving while the ones at the top are moving very quickly. Without the formula above, one would have to use an integral to add the momenta of all of the mass elements to find the total momentum of the disk. However, the formula above tells us that we can simply multiply the mass of the disk by the velocity of its center of mass, which is in this case the geometric center of the disk. This reveals that the fact that the disk is rotating does not affect its momentum.
==Worked Example 2: Changing Speed==


====Impulse====
A car has momentum of <math>20,000 kg \cdot m/s</math>. How does the momentum change if the car slows to half its original speed?
Impulse is the change in momentum. It is the integral of force over a time interval. Therefore, impulse equals to the force, F, multiplied by the change in time, t. The unit for impulse is newton*second. Since impulse is the change in momentum, an alternative formula of impulse is mass, m, multiplied by change in velocity, v. Both the formulae mentioned are important to remember to solve impulse-related questions.


====In Relation to Other Physics Topics====
Momentum is:


In the absence of a net force, the momentum of a particle stays constant over time, as stated by Newton's first law.
<math>p = mv</math>


When a force is applied to a particle, its momentum evolves over time according to Newton's second law. For more information, see [[Newton's Second Law: the Momentum Principle]].
If the mass stays the same but the velocity is cut in half, then the momentum is also cut in half.


When an impulse is applied to a particle, its momentum changes in a specific way: the change in momentum <math>\Delta\vec{p}</math> is equal to the impulse <math>\vec{J}</math>. This is a consequence of the Momentum Principle. For more information, see [[Impulse and Momentum]].
<math>p_{new} = \frac{1}{2}p</math>


When the net external force on a system of particles is 0, the system's momentum is conserved (that is, constant over time) even if the particles within the system interact with each other (exert internal forces on each other). For more information, see [[Conservation of Momentum]].
<math>p_{new} = \frac{1}{2}(20,000)</math>


===A Computational Model===
<math>p_{new} = 10,000 kg \cdot m/s</math>


In computational simulations of particles using [[Iterative Prediction]], a momentum vector variable is assigned to each particle. Such simulations usually in "time steps," or iterations of a loop representing a short time interval. In each time step, the particles' momenta are updated based on the forces acting on it. Then their velocities are calculated by dividing each particle's momentum by its mass. Finally, the velocities are used to update the positions of the particles. Below is an example of such a simulation:
Final answer:


This vPython simulation shows a cart (represented by a rectangle) whose motion is affected by a gust of wind applying a constant force.
<math>\boxed{10,000 kg \cdot m/s}</math>


https://trinket.io/glowscript/ce43925647
==Worked Example 3: Finding Mass from Change in Momentum==


For more information, see [[Iterative Prediction]].
A basketball is rolling to the right at 23.5 m/s. Later, it is still rolling to the right but has slowed to 3.8 m/s. The change in momentum is <math>-17.24 kg \cdot m/s</math>. What is the mass of the basketball?


==Examples==
Let right be the positive direction.


===1. (Simple)===
Given:
Find the momentum of a ball that has a mass of 69kg and is moving at <1,2,3> m/s.


[[File:momentumsimple.jpg]]
<math>v_i = 23.5 m/s</math>


===2. (Middling)===
<math>v_f = 3.8 m/s</math>
A car has 20,000 N of momentum.
How would the momentum of the car change if:
a) the car slowed to half of its speed?
b) the car completely stopped?
c) the car gained its original weight in luggage?


[[File:momentummiddling.jpg]]
<math>\Delta p = -17.24 kg \cdot m/s</math>


===3. (Difficult)===
Use:
You and your friends are watching NBA highlights at home and want to practice your physics. You notice at the beginning of a clip a basketball ball is rolling down the court at 23.5 m/s to the right. At the end, it is rolling at 3.8 m/s in the same direction. The commentator tells you that the change in its momentum is 17.24 kg m/s to the left. Curious at how many basketballs you can carry, you want to find the mass of the ball.


[[File:momentumhardaf.jpg]]
<math>\Delta p = m(v_f - v_i)</math>


===4. (Difficult)===
Substitute:
A system is comprised of two particles. One particle has a mass of 6kg and is travelling in a direction 30 degrees east of north at a speed of 8m/s. The total momentum of the system is 3kg*m/s west. The second particle has a mass of 3kg. What is the second particle's velocity? Give your answer in terms of a north-south component and an east-west component.


[[File:Momentumadditionalhardsmaller.jpg]]
<math>-17.24 = m(3.8 - 23.5)</math>


==Connectedness==
<math>-17.24 = m(-19.7)</math>


===Scenario: runaway vehicle===
Solve for mass:


Imagine that you are standing at the bottom of a hill when a runaway vehicle comes careening down. If it is a bicycle, it would be much easier to stop than if it were a truck moving at the same speed. One explanation for this is that the truck would have a greater mass and therefore a greater momentum. In order to be brought to rest, the truck must therefore experience a large change in momentum, which means a large impulse must be exerted on it.
<math>m = \frac{-17.24}{-19.7}</math>


==History==
<math>m = 0.875 kg</math>


The oldest known attempt at quantifying motion using both an object's speed and mass was that of René Descartes (1596–1650) [https://science.jrank.org/pages/4419/Momentum.html (source)]. John Wallis (1616 – 1703) was the first to use the phrase momentum, and to define it as the product of mass and velocity (rather than speed). He recognized that this notion of momentum is conserved in closed systems [https://www.famousscientists.org/john-wallis/ (source)], a concept confirmed by experiments performed by Christian Huygens (1629-1695) [https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html (source)]. Finally, Isaac Newton (1643-1727) wrote his famous three laws relating momentum to force in his book Principia Mathematica in 1687. These offered a theoretical explanation for conservation of momentum. These foundations were so logically sound and experimentally observable that they would go unquestioned for centuries, until Albert Einstein (1879-1955) had to adjust the formula for momentum to maintain its accuracy for objects moving at relativistic speeds (see [[relativistic momentum]]).
Final answer:


== See also ==
<math>\boxed{m \approx 0.875 kg}</math>


*[[Mass]]
==Worked Example 4: Two-Particle System==
*[[Velocity]]
*[[Vectors]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Conservation of Momentum]]


===Further reading===
A system is made of two particles. Particle 1 has a mass of 6 kg and moves at 8 m/s at 30 degrees east of north. The total momentum of the system is 3 kg·m/s west. Particle 2 has a mass of 3 kg. Find the velocity components of particle 2.


Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.
First find the velocity components of particle 1.


===External links===
North component:


<ol>
<math>v_{1N} = 8\cos(30^\circ)</math>
<li>https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum</li>
 
</ol>
<math>v_{1N} = 6.93 m/s</math>
 
East component:
 
<math>v_{1E} = 8\sin(30^\circ)</math>
 
<math>v_{1E} = 4.00 m/s</math>
 
Now find the momentum components of particle 1.
 
North momentum:
 
<math>p_{1N} = 6(6.93)</math>
 
<math>p_{1N} = 41.58 kg \cdot m/s</math>
 
East momentum:
 
<math>p_{1E} = 6(4.00)</math>
 
<math>p_{1E} = 24.00 kg \cdot m/s</math>
 
The total momentum is 3 kg·m/s west, so:
 
<math>p_{total,N} = 0</math>
 
<math>p_{total,E} = -3 kg \cdot m/s</math>
 
Find particle 2's momentum.
 
North:
 
<math>0 = 41.58 + p_{2N}</math>
 
<math>p_{2N} = -41.58 kg \cdot m/s</math>
 
East:
 
<math>-3 = 24 + p_{2E}</math>
 
<math>p_{2E} = -27 kg \cdot m/s</math>
 
Now divide by the mass of particle 2.
 
North velocity:
 
<math>v_{2N} = \frac{-41.58}{3}</math>
 
<math>v_{2N} = -13.86 m/s</math>
 
East velocity:
 
<math>v_{2E} = \frac{-27}{3}</math>
 
<math>v_{2E} = -9.00 m/s</math>
 
Final answer:
 
Particle 2 moves at:
 
<math>\boxed{13.86 m/s \text{ south and } 9.00 m/s \text{ west}}</math>
 
==Computational Model==
 
In a computational model, momentum can be updated using small time steps. If the net force on an object is known, the momentum can be updated with:
 
<pre>
p = p + Fnet*deltat
</pre>
 
After updating momentum, the velocity can be found using:
 
<pre>
v = p/m
</pre>
 
Then the position can be updated using:
 
<pre>
r = r + v*deltat
</pre>
 
This method is useful for simulations because the computer repeats these steps many times.
 
A simple GlowScript simulation for this topic could show a ball moving across the screen. A force could act on the ball for a short amount of time, changing its momentum. The simulation could display the momentum vector before and after the force acts.
 
==Real-World Applications==
 
===Runaway Vehicle===
 
A runaway truck is harder to stop than a bicycle moving at the same speed because the truck has much more mass. Since momentum depends on mass and velocity, the truck has more momentum. To stop the truck, a much larger change in momentum is needed.
 
===Sports===
 
Momentum is important in sports. A fast-moving football player, baseball, or basketball has momentum. To stop or redirect the object, a force must change its momentum.
 
===Car Crashes===
 
In a car crash, the car and passengers experience a large change in momentum. Safety features such as airbags, seatbelts, and crumple zones help increase the time over which momentum changes. This reduces the average force on passengers.
 
===Rocket Motion===
 
Rockets use momentum conservation to move. Exhaust gases are pushed backward, and the rocket moves forward. This is why rockets can accelerate even in outer space.
 
==Common Mistakes==
 
* Using newtons as the unit for momentum.
* Forgetting that momentum is a vector.
* Forgetting to include direction.
* Confusing mass and weight.
* Thinking a heavier object always has more momentum, even if it is not moving.
* Forgetting that an object at rest has zero momentum.
* Adding momenta like regular numbers instead of vector components.
 
==Practice Questions==
 
# A 5 kg object moves with velocity <math>\langle 3,0,0 \rangle m/s</math>. What is its momentum?
# A 2 kg object has velocity <math>\langle -4,2,0 \rangle m/s</math>. Find its momentum.
# A car's momentum is cut in half while its mass stays the same. What happened to its velocity?
# Two carts have momenta <math>\langle 10,0,0 \rangle kg \cdot m/s</math> and <math>\langle -6,0,0 \rangle kg \cdot m/s</math>. What is the total momentum of the system?
# A 0.5 kg ball changes velocity from 12 m/s to 4 m/s. What is its change in momentum?
 
==See Also==
 
* [[Mass]]
* [[Velocity]]
* [[Vectors]]
* [[Newton's Second Law: the Momentum Principle]]
* [[Impulse and Momentum]]
* [[Conservation of Momentum]]
* [[Collisions]]
* [[Center of Mass]]


==References==
==References==


 
* OpenStax. ''College Physics 2e'', Linear Momentum and Collisions.
[[Category: Momentum]]
* Khan Academy. ''Introduction to Momentum.''
*http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
* HyperPhysics. ''Momentum.''
*http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.html
* The Physics Classroom. ''Momentum and Impulse Connection.''
*https://science.jrank.org/pages/4419/Momentum.html
*https://www.famousscientists.org/john-wallis/
*https://www2.stetson.edu/~efriedma/periodictable/html/Hg.html

Latest revision as of 16:30, 28 April 2026

Edwyn Torres — Spring 2026

Linear Momentum

Introduction

Linear momentum is a vector quantity that describes the motion of an object. Momentum depends on two things: the object's mass and the object's velocity. An object with more mass or more velocity has more momentum.

Momentum is useful because it helps describe how difficult it is to stop or change the motion of an object. For example, a large truck and a small bicycle can move at the same speed, but the truck has much more momentum because it has much more mass.

The Main Idea

The linear momentum of an object is defined as:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

where:

  • [math]\displaystyle{ \vec{p} }[/math] = linear momentum
  • [math]\displaystyle{ m }[/math] = mass
  • [math]\displaystyle{ \vec{v} }[/math] = velocity

Since velocity is a vector, momentum is also a vector. This means momentum has both magnitude and direction. The direction of an object's momentum is the same as the direction of its velocity.

Key Equations

For one object:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

For a system of multiple objects:

[math]\displaystyle{ \vec{p}_{total} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + ... }[/math]

For momentum change:

[math]\displaystyle{ \Delta \vec{p} = \vec{p}_f - \vec{p}_i }[/math]

Impulse-momentum relationship:

[math]\displaystyle{ \Delta \vec{p} = \vec{F}_{net}\Delta t }[/math]

or:

[math]\displaystyle{ \vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t }[/math]

Units

The SI unit for momentum is:

[math]\displaystyle{ kg \cdot m/s }[/math]

Momentum should not be measured in newtons. A newton is a unit of force:

[math]\displaystyle{ 1N = 1kg \cdot m/s^2 }[/math]

Impulse has units of:

[math]\displaystyle{ N \cdot s }[/math]

Since:

[math]\displaystyle{ N \cdot s = kg \cdot m/s }[/math]

impulse and momentum have equivalent units.

Conceptual Explanation

Momentum describes how much motion an object has. If two objects have the same velocity, the object with more mass has more momentum. If two objects have the same mass, the object with greater velocity has more momentum.

For example, a bowling ball rolling at 3 m/s has more momentum than a tennis ball rolling at 3 m/s because the bowling ball has more mass. A fast-moving baseball can also have a large amount of momentum because its velocity is high, even though its mass is small.

Momentum is especially important in collisions. During a collision, objects can exert large forces on each other over a short time interval. Even if the forces are complicated, momentum can still be used to analyze the motion before and after the collision.

Single Particle Momentum

For a single particle, momentum is calculated using:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

Example:

If a 4 kg object moves with velocity [math]\displaystyle{ \langle 2, 0, 0 \rangle m/s }[/math], then its momentum is:

[math]\displaystyle{ \vec{p} = 4\langle 2,0,0 \rangle }[/math]

[math]\displaystyle{ \vec{p} = \langle 8,0,0 \rangle kg \cdot m/s }[/math]

This means the object has momentum in the positive x-direction.

Multiple Particle Systems

For a system with more than one object, the total momentum is found by adding the momentum vectors of each object.

[math]\displaystyle{ \vec{p}_{total} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + ... }[/math]

Because momentum is a vector, directions must be included. Momentum in opposite directions can cancel.

For example, if one object has momentum:

[math]\displaystyle{ \vec{p}_1 = \langle 10,0,0 \rangle kg \cdot m/s }[/math]

and another object has momentum:

[math]\displaystyle{ \vec{p}_2 = \langle -4,0,0 \rangle kg \cdot m/s }[/math]

then the total momentum is:

[math]\displaystyle{ \vec{p}_{total} = \langle 10,0,0 \rangle + \langle -4,0,0 \rangle }[/math]

[math]\displaystyle{ \vec{p}_{total} = \langle 6,0,0 \rangle kg \cdot m/s }[/math]

The system still has momentum in the positive x-direction.

Momentum and Impulse

Momentum changes when a net force acts on an object for a period of time. This change in momentum is called impulse.

[math]\displaystyle{ \Delta \vec{p} = \vec{F}_{net}\Delta t }[/math]

A larger force creates a larger change in momentum. A longer time interval also creates a larger change in momentum.

This explains why airbags help protect people during car crashes. In a crash, a person's momentum must decrease to zero. An airbag increases the time over which the person slows down. Since the stopping time is longer, the average force on the person is smaller.

Momentum and Newton's Laws

Momentum is closely connected to Newton's laws.

Newton's First Law says that an object will continue moving at constant velocity unless acted on by a net external force. In terms of momentum, this means an object's momentum remains constant if the net force is zero.

Newton's Second Law can be written using momentum:

[math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]

This means the net force on an object is equal to the rate of change of its momentum.

Momentum in Rocket Propulsion

Rockets move because of conservation of momentum. A rocket pushes exhaust gases backward at high speed. Because the gases gain momentum backward, the rocket gains momentum forward.

This works even in space because the rocket does not need to push against air. Instead, the rocket changes its momentum by ejecting mass in the opposite direction.

Common Misconceptions

Misconception 1: Momentum and velocity are the same thing

Momentum and velocity are related, but they are not the same. Momentum depends on both mass and velocity. A heavy object moving slowly can have the same momentum as a light object moving quickly.

Misconception 2: Momentum is measured in newtons

Momentum is not measured in newtons. Momentum is measured in:

[math]\displaystyle{ kg \cdot m/s }[/math]

Newtons are used for force.

Misconception 3: Direction does not matter

Momentum is a vector, so direction always matters. A positive momentum and a negative momentum can partially or completely cancel when finding total momentum.

Misconception 4: Momentum only matters in collisions

Momentum is very important in collisions, but it also matters in rocket motion, sports, walking, running, explosions, and systems of particles.

Worked Example 1: Basic Momentum

A 70 kg person is moving with velocity [math]\displaystyle{ \langle 1,2,3 \rangle m/s }[/math]. What is the person's momentum?

Use:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

Substitute:

[math]\displaystyle{ \vec{p} = 70\langle 1,2,3 \rangle }[/math]

Multiply each component by 70:

[math]\displaystyle{ \vec{p} = \langle 70,140,210 \rangle kg \cdot m/s }[/math]

Final answer:

[math]\displaystyle{ \boxed{\vec{p} = \langle 70,140,210 \rangle kg \cdot m/s} }[/math]

Worked Example 2: Changing Speed

A car has momentum of [math]\displaystyle{ 20,000 kg \cdot m/s }[/math]. How does the momentum change if the car slows to half its original speed?

Momentum is:

[math]\displaystyle{ p = mv }[/math]

If the mass stays the same but the velocity is cut in half, then the momentum is also cut in half.

[math]\displaystyle{ p_{new} = \frac{1}{2}p }[/math]

[math]\displaystyle{ p_{new} = \frac{1}{2}(20,000) }[/math]

[math]\displaystyle{ p_{new} = 10,000 kg \cdot m/s }[/math]

Final answer:

[math]\displaystyle{ \boxed{10,000 kg \cdot m/s} }[/math]

Worked Example 3: Finding Mass from Change in Momentum

A basketball is rolling to the right at 23.5 m/s. Later, it is still rolling to the right but has slowed to 3.8 m/s. The change in momentum is [math]\displaystyle{ -17.24 kg \cdot m/s }[/math]. What is the mass of the basketball?

Let right be the positive direction.

Given:

[math]\displaystyle{ v_i = 23.5 m/s }[/math]

[math]\displaystyle{ v_f = 3.8 m/s }[/math]

[math]\displaystyle{ \Delta p = -17.24 kg \cdot m/s }[/math]

Use:

[math]\displaystyle{ \Delta p = m(v_f - v_i) }[/math]

Substitute:

[math]\displaystyle{ -17.24 = m(3.8 - 23.5) }[/math]

[math]\displaystyle{ -17.24 = m(-19.7) }[/math]

Solve for mass:

[math]\displaystyle{ m = \frac{-17.24}{-19.7} }[/math]

[math]\displaystyle{ m = 0.875 kg }[/math]

Final answer:

[math]\displaystyle{ \boxed{m \approx 0.875 kg} }[/math]

Worked Example 4: Two-Particle System

A system is made of two particles. Particle 1 has a mass of 6 kg and moves at 8 m/s at 30 degrees east of north. The total momentum of the system is 3 kg·m/s west. Particle 2 has a mass of 3 kg. Find the velocity components of particle 2.

First find the velocity components of particle 1.

North component:

[math]\displaystyle{ v_{1N} = 8\cos(30^\circ) }[/math]

[math]\displaystyle{ v_{1N} = 6.93 m/s }[/math]

East component:

[math]\displaystyle{ v_{1E} = 8\sin(30^\circ) }[/math]

[math]\displaystyle{ v_{1E} = 4.00 m/s }[/math]

Now find the momentum components of particle 1.

North momentum:

[math]\displaystyle{ p_{1N} = 6(6.93) }[/math]

[math]\displaystyle{ p_{1N} = 41.58 kg \cdot m/s }[/math]

East momentum:

[math]\displaystyle{ p_{1E} = 6(4.00) }[/math]

[math]\displaystyle{ p_{1E} = 24.00 kg \cdot m/s }[/math]

The total momentum is 3 kg·m/s west, so:

[math]\displaystyle{ p_{total,N} = 0 }[/math]

[math]\displaystyle{ p_{total,E} = -3 kg \cdot m/s }[/math]

Find particle 2's momentum.

North:

[math]\displaystyle{ 0 = 41.58 + p_{2N} }[/math]

[math]\displaystyle{ p_{2N} = -41.58 kg \cdot m/s }[/math]

East:

[math]\displaystyle{ -3 = 24 + p_{2E} }[/math]

[math]\displaystyle{ p_{2E} = -27 kg \cdot m/s }[/math]

Now divide by the mass of particle 2.

North velocity:

[math]\displaystyle{ v_{2N} = \frac{-41.58}{3} }[/math]

[math]\displaystyle{ v_{2N} = -13.86 m/s }[/math]

East velocity:

[math]\displaystyle{ v_{2E} = \frac{-27}{3} }[/math]

[math]\displaystyle{ v_{2E} = -9.00 m/s }[/math]

Final answer:

Particle 2 moves at:

[math]\displaystyle{ \boxed{13.86 m/s \text{ south and } 9.00 m/s \text{ west}} }[/math]

Computational Model

In a computational model, momentum can be updated using small time steps. If the net force on an object is known, the momentum can be updated with: 

p = p + Fnet*deltat

After updating momentum, the velocity can be found using: 

v = p/m

Then the position can be updated using: 

r = r + v*deltat

This method is useful for simulations because the computer repeats these steps many times.

A simple GlowScript simulation for this topic could show a ball moving across the screen. A force could act on the ball for a short amount of time, changing its momentum. The simulation could display the momentum vector before and after the force acts.

Real-World Applications

Runaway Vehicle

A runaway truck is harder to stop than a bicycle moving at the same speed because the truck has much more mass. Since momentum depends on mass and velocity, the truck has more momentum. To stop the truck, a much larger change in momentum is needed.

Sports

Momentum is important in sports. A fast-moving football player, baseball, or basketball has momentum. To stop or redirect the object, a force must change its momentum.

Car Crashes

In a car crash, the car and passengers experience a large change in momentum. Safety features such as airbags, seatbelts, and crumple zones help increase the time over which momentum changes. This reduces the average force on passengers.

Rocket Motion

Rockets use momentum conservation to move. Exhaust gases are pushed backward, and the rocket moves forward. This is why rockets can accelerate even in outer space.

Common Mistakes

  • Using newtons as the unit for momentum.
  • Forgetting that momentum is a vector.
  • Forgetting to include direction.
  • Confusing mass and weight.
  • Thinking a heavier object always has more momentum, even if it is not moving.
  • Forgetting that an object at rest has zero momentum.
  • Adding momenta like regular numbers instead of vector components.

Practice Questions

  1. A 5 kg object moves with velocity [math]\displaystyle{ \langle 3,0,0 \rangle m/s }[/math]. What is its momentum?
  2. A 2 kg object has velocity [math]\displaystyle{ \langle -4,2,0 \rangle m/s }[/math]. Find its momentum.
  3. A car's momentum is cut in half while its mass stays the same. What happened to its velocity?
  4. Two carts have momenta [math]\displaystyle{ \langle 10,0,0 \rangle kg \cdot m/s }[/math] and [math]\displaystyle{ \langle -6,0,0 \rangle kg \cdot m/s }[/math]. What is the total momentum of the system?
  5. A 0.5 kg ball changes velocity from 12 m/s to 4 m/s. What is its change in momentum?

See Also

References

  • OpenStax. College Physics 2e, Linear Momentum and Collisions.
  • Khan Academy. Introduction to Momentum.
  • HyperPhysics. Momentum.
  • The Physics Classroom. Momentum and Impulse Connection.
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