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This page defines the linear momentum of a particle or system.
<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 represented by the letter <math>\vec{p}</math> and is often referred to as simply "momentum."
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.


===A Mathematical Model===
For example, if one object has momentum:


====Single Particles====
<math>\vec{p}_1 = \langle 10,0,0 \rangle kg \cdot m/s</math>
Linear momentum is a vector quantity, like  velocity, possessing a direction as well as a magnitude:
:<math>\mathbf{p} = m \mathbf{v}</math>
where p is the vector stating the object's momentum in the three directions of space, and where v is the three-dimensional velocity vector giving the object's movement in each of these directions, and m is the object's mass.


====Multiple Particles====
and another object has momentum:


The momentum of a system of particles is the sum of both particles' momentum (momenta). If the particles have masses m<sub>1</sub> and m<sub>2</sub>, respectively, and velocities v<sub>1</sub> and v<sub>2</sub>, the total momentum is:
<math>\vec{p}_2 = \langle -4,0,0 \rangle kg \cdot m/s</math>
:<math> \begin{align} p &= p_1 + p_2 \\
&= m_1 v_1 + m_2 v_2\,. \end{align} </math>
This model can be used to measure the momentum of a system of any amount of particles.


A system of particles has a [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html center of mass], which is a point determined by the weighted sum of each of their positions:
then the total momentum is:
:<math> r_\text{cm} = \frac{m_1 r_1 + m_2 r_2 + \cdots}{m_1 + m_2 + \cdots} = \frac{\sum\limits_{i}m_ir_i}{\sum\limits_{i}m_i}.</math>


If all the particles are in motion, the center of mass will most likely be moving as well (unless the particles are in rotation around it). If the center of mass is moving at velocity v<sub>cm</sub>, the momentum can be found by using:
<math>\vec{p}_{total} = \langle 10,0,0 \rangle + \langle -4,0,0 \rangle</math>
:<math>p= mv_\text{cm}.</math>


====In Relation to Newton's Second Law====
<math>\vec{p}_{total} = \langle 6,0,0 \rangle kg \cdot m/s</math>


When a force (<math>F</math>) is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount
The system still has momentum in the positive x-direction.
:<math>\Delta p = F \Delta t\,.</math>
This quantity is called [http://www.sparknotes.com/testprep/books/sat2/physics/chapter9section2.rhtml impulse].


When differentiated, this represents [http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law Newton's Second Law]. Put simply, rate of change of a particle's momentum is proportional to the force <math>F</math> acting on it over a certain interval of time. This is symbolized by
==Momentum and Impulse==
:<math>F = \frac{dp }{d t}. </math>


===A Computational Model===
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>


In this code I have simulated what happens when a cart (visualized by the box) is acted upon by the force of a small gust of wind.  
A larger force creates a larger change in momentum. A longer time interval also creates a larger change in momentum.
Its initial velocity and mass is defined and you can clearly see it being affected by the wind.
Due to the wind, its momentum changes because its velocity (both direction and speed) is affected by the wind.
https://trinket.io/glowscript/ce43925647


==Examples==
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.


===Scenario===
==Momentum and Newton's Laws==


If you are standing at the bottom of a hill and faced with the option of stopping a runaway semi-truck or a runaway bicycle, you would probably choose to stop the bike because the semi truck has more momentum. Momentum simply means mass in motion. The semi truck has larger momentum because it is massive, but also has more speed.
Momentum is closely connected to Newton's laws.


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


[[File:momentumsimple.jpg]]
Newton's Second Law can be written using momentum:


===Middling===
<math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</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?


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


[[File:momentummiddling.jpg]]
==Momentum in Rocket Propulsion==


===Difficult===
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.
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]]
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.


==Connectedness==
==Common Misconceptions==


===How is momentum used in Electrical Engineering?===
===Misconception 1: Momentum and velocity are the same thing===
Electrokinetic momentum is used in many devices to calculate the voltage necessary to change the current in an inductive circuit. This calculation is utilized within many electrical devices such as resistance grids.


==History==
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.


===Newton's 2nd Law===
===Misconception 2: Momentum is measured in newtons===


While Newton's 1st Law was not entirely his own, his 2nd and 3rd are.
Momentum is not measured in newtons. Momentum is measured in:


Original Latin:
<math>kg \cdot m/s</math>


“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”
Newtons are used for force.


This was translated closely in Motte's 1729:
===Misconception 3: Direction does not matter===


“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”
Momentum is a vector, so direction always matters. A positive momentum and a negative momentum can partially or completely cancel when finding total momentum.


Essentially:
===Misconception 4: Momentum only matters in collisions===


The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed. In other words, F=ma.
Momentum is very important in collisions, but it also matters in rocket motion, sports, walking, running, explosions, and systems of particles.


== See also ==
==Worked Example 1: Basic Momentum==


*http://www.physicsbook.gatech.edu/Velocity
A 70 kg person is moving with velocity <math>\langle 1,2,3 \rangle m/s</math>. What is the person's momentum?
*http://www.physicsbook.gatech.edu/Mass
*http://www.physicsbook.gatech.edu/Vectors
*http://www.physicsbook.gatech.edu/Newton%E2%80%99s_Second_Law_of_Motion


===Further reading===
Use:


Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 1). Raleigh, North Carolina: Wiley.
<math>\vec{p} = m\vec{v}</math>
 
Substitute:
 
<math>\vec{p} = 70\langle 1,2,3 \rangle</math>
 
Multiply each component by 70:
 
<math>\vec{p} = \langle 70,140,210 \rangle kg \cdot m/s</math>
 
Final answer:
 
<math>\boxed{\vec{p} = \langle 70,140,210 \rangle kg \cdot m/s}</math>
 
==Worked Example 2: Changing Speed==
 
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?
 
Momentum is:
 
<math>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>p_{new} = \frac{1}{2}p</math>
 
<math>p_{new} = \frac{1}{2}(20,000)</math>
 
<math>p_{new} = 10,000 kg \cdot m/s</math>
 
Final answer:
 
<math>\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>-17.24 kg \cdot m/s</math>. What is the mass of the basketball?
 
Let right be the positive direction.
 
Given:
 
<math>v_i = 23.5 m/s</math>
 
<math>v_f = 3.8 m/s</math>
 
<math>\Delta p = -17.24 kg \cdot m/s</math>
 
Use:
 
<math>\Delta p = m(v_f - v_i)</math>
 
Substitute:
 
<math>-17.24 = m(3.8 - 23.5)</math>
 
<math>-17.24 = m(-19.7)</math>
 
Solve for mass:
 
<math>m = \frac{-17.24}{-19.7}</math>
 
<math>m = 0.875 kg</math>
 
Final answer:
 
<math>\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>v_{1N} = 8\cos(30^\circ)</math>
 
<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:Which Category did you place this in?]]
* Khan Academy. ''Introduction to Momentum.''
*http://www.sparknotes.com/testprep/books/sat2/physics/chapter9section1.rhtml
* HyperPhysics. ''Momentum.''
*http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
* The Physics Classroom. ''Momentum and Impulse Connection.''
*http://www.sparknotes.com/testprep/books/sat2/physics/chapter9section2.rhtml
*http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
*http://study.com/academy/lesson/linear-momentum-definition-equation-and-examples.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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