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==Impulse Momentum==
<big><b>Edwyn Torres — Spring 2026</b></big>


This topic focuses on the simple relation between momentum and impulse.
=Impulse and Momentum=


===Impulse Momentum Theorem===
==Introduction==


The Impulse Momentum Theorem relates the momentum of a body or system to the force acting on the body. Impulse(J) is also the change in momentum. As a force on a body is applied for a longer amount of time, the impulse also changes. If there is a changing force over the same time interval, the impulse also changes. The impulse is the product of the average force and the time interval over which it acts. Like linear momentum, impulse is a vector quantity and has the same direction as the average force. Its units are given in Newton-seconds (Ns).  
Impulse and momentum are important ideas in physics because they help explain how forces change the motion of objects. Momentum describes how difficult it is to stop or change the motion of an object. Impulse describes how much a force changes an object's momentum over a period of time.


A large impulse will cause a large change in an object's momentum, just as a small impulse will cause a smaller change in an object's momentum. When looking at the equation <math>{J} = {d\vec{p}}</math>, one can replace J with the product of the average force and the time interval. Rearranging that equation results in <math>{F} = {\frac{d\vec{p}}{dt}}</math>, which shows that whenever momentum changes with time, there is some force acting on the body.  
A large force acting for a short time can create the same impulse as a smaller force acting for a longer time. This idea is important in collisions, sports, car safety, and many other real-world situations.


To clarify, impulse is the effect of a net force acting on a body over a period of time, while momentum is the force within a body or system due to its total velocity.
==Main Idea==


===Real World Applications===
Momentum is a vector quantity that depends on mass and velocity:
To help understand the importance of the Impulse Momentum Theorem and how it can be used to determine the effect of a force over time, check out these real world examples:  


This video demonstrates what happens when dropping an egg on a hard surface versus dropping an egg on a cushioned surface. The cushioning effect increases the time of the collision with the egg, resulting in less splatter: [https://youtu.be/B-_3vi3eDb8?t=50s]
<math>\vec{p} = m\vec{v}</math>


Watch a car collide with a wall and how the velocity, mass, and the time of the collision all play a role in determining the magnitude of force in the accident. [https://youtu.be/B-_3vi3eDb8?t=50s]
Impulse is also a vector quantity. It is the effect of a net force acting over a time interval:


The same concept applies when punching someone in the face. [[Media:u4l1c7.gif]] To create a larger force and make the punch hurt more, the blow should be applied swiftly, because the decreased collision time increases the amount of force. This is why many boxers have learned to protect themselves from more serious injury by relaxing their necks and letting their heads move with a punch to the head if they can not block it in time. Allowing the head to move backwards upon impact increases the time of the impact of the glove with their head and therefore minimizing the effect of the force.
<math>\vec{J} = \vec{F}_{net,avg}\Delta t</math>


====A Mathematical Model====
The impulse-momentum theorem says that impulse equals the change in momentum:
Impulse can mathematically be defined as the force on a body multiplied by the duration of that force.
<math>{\frac{d\vec{p}}{dt}} = \vec{F}_{net} = {m}{\frac{d\vec{v}}{dt}}</math> where '''p''' is the momentum of the system and '''F''' is the net force, '''m''' is the mass, and '''v''' is velocity.
This can be rearranged to represent impulse, '''J''' as a relationship between the net force and time of the collision:  
<math>{J} = {d\vec{p}} = \vec{F}_{net}{dt}</math>


====A Computational Model====
<math>\vec{J} = \Delta \vec{p}</math>
It is possible to predict location and velocity by using the momentum and impulse principles. Here is an example of code that will run in VPython (not GlowScript) which should show a fancart moving at a specified velocity and how the momentum changes over time as the force of air is applied to it.


<code>
This can also be written as:
  from __future__ import division
  from visual import *
  #Create a track on which to drive a cart
  track = box(pos=vector(0,-.05,0), size=(2.0,0.05,.10), color=color.white)
  #Create cart object as a smaller box than the track.
  cart = box(pos=vector(0.046,0,0), size=(0.1,0.04,0.06), color=color.blue)
  #Set initial conditions         
  mcart=0.4813
  vcart=vector(0.429,0.109,0)
  #Record momentum equation
  pcart=mcart*vcart
  #Record the time interval
  deltat=0.01
  t=0
  #Account for the main force acting on the cart: the air resistance slowing it down.
  Fair=vector(0.175,0,0)
  #Run the cart for a certain amount of time to see how the momentum changes.
  while t<=4.81:
    #Use impulse equation to update momentum, since F*deltat is equal to the change in momentum.
    #Updates momentum for every iteration, thus using impulse to keep track of the change.
    pcart=pcart+Fair*deltat
    #Update position of the cart.
    cart.pos=cart.pos+(pcart/mcart)*deltat
    vcart=pcart/mcart
    #Update the time, to keep track of the total time and thus number of iteration.
    t=t+deltat
    rate(100)
  #See the numerical results.
  print("after the loop")
  print(cart.pos)
  print("cart momentum=", pcart)
  print("cart position=", cart.pos)
  print("cart velocity=", vcart)


</code>
<math>\vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t</math>


==Examples==
This equation is useful because it lets us predict how an object's momentum changes when a force acts on it.


Be sure to show all steps in your solution and include diagrams whenever possible
==Key Equations==


===Simple===
* <math>\vec{p} = m\vec{v}</math>
Q: What is the impulse imparted by a rocket that exerts 4.8 N for 1.63 seconds?
* <math>\vec{J} = \vec{F}_{net,avg}\Delta t</math>
* <math>\vec{J} = \Delta \vec{p}</math>
* <math>\Delta \vec{p} = \vec{p}_f - \vec{p}_i</math>
* <math>\vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t</math>


A: As we know, <math>{J} = {d\vec{p}} = \vec{F}_{net}{dt}</math>. From the given information, we can conclude that <math>{F} = {4.8N}</math> and <math>{dt} = {1.63 seconds}</math> therefore, <math>{J} = {4.8} \cdot{1.63} = {7.8 Ns}</math>
Where:


===Middling===
* <math>\vec{p}</math> = momentum
This physics teacher works out a slightly more involved impulse collision analysis on a real car crash from a safety video. Follow along here: [https://youtu.be/ph48Xwj_eS8?t=3m20s]
* <math>m</math> = mass
* <math>\vec{v}</math> = velocity
* <math>\vec{J}</math> = impulse
* <math>\vec{F}_{net}</math> = net force
* <math>\Delta t</math> = time interval


===Difficult===
==Units==
For you engineers, here is a more advanced impulse problem using calculus to solve a mechanics problem: [https://www.youtube.com/watch?v=avmaZF5AY4I]
==Connectedness==
One way to think of the importance of measuring impulse and its relationship to force is by imagining a car collision. When a car hits, for example, a wall or another car, a certain amount of force from the impact will cause the airbags in the car, ultimately leading to fewer deaths and injuries among drivers and passengers than if there had not been an air bag. When you think about it, due to the relationship between impulse, force, and time duration, the force of a collision where the momentum is changing is indirectly proportional to the time interval over which the force acts. To decrease the force, the time before the final impact must be increased. An air bag fulfills that necessity by inflating and thus creating a barrier between the human inside of the car and the other car or object with which it collides. The airbag provides an opposing force over an interval of time. The inflated airbag reduces the speed of the human and increases the time before the person in the car comes to a full stop because of the collision. As a result, the impulsive force decreases, ultimately creating a safer impact for all passengers.


Another instance is when a person jumps off of something and lands on his or her feet. When jumping off of a raised surface and landing on one's feet, a person's knees naturally bend. This bending of the knees is essential for minimizing the amount of force from the impact that actually reaches a human's knees. When landing, the bent knees causes the collision between one's feet and the ground to last longer, and as the time of the collision increases, the force decreases because they are indirectly related.If a person didn't bend his knees, there would be serious damage done to the bones and joints. 
The unit for momentum is:


==History==
<math>kg \cdot m/s</math>


There is no specific record as to who discovered the Impulse-Momentum Theorem. However, it is derived from and logically correlates to Newton's Second Law, which states that the net force of an object is directly related to the rate of change of its linear momentum: <math>{\frac{d\vec{p}}{dt}} = \vec{F}_{net}</math>.
The unit for impulse is:


== See also ==
<math>N \cdot s</math>


A great video that explains this concept with visual examples: [https://www.youtube.com/watch?v=ph48Xwj_eS8]
These units are equivalent because:


[http://www.physicsbook.gatech.edu/Momentum_Principle]
<math>1N = 1kg \cdot m/s^2</math>


===Further reading===
So:


RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.
<math>N \cdot s = kg \cdot m/s</math>


===External links===
This makes sense because impulse equals change in momentum.


[http://www.physicsclassroom.com/class/momentum/Lesson-1/Momentum-and-Impulse-Connection
==Conceptual Explanation==
]
 
Impulse depends on two things: force and time. If the force is larger, the impulse is larger. If the force acts for a longer time, the impulse is also larger.
 
For example, when catching a baseball, a person usually moves their hand backward as the ball arrives. This increases the time over which the ball slows down. Since the ball still needs the same change in momentum to stop, increasing the stopping time decreases the average force on the hand.
 
This is why catching a ball with stiff hands hurts more than catching it while moving your hands backward.
 
==Worked Example 1: Finding Final Momentum==
 
A 2 kg cart is moving with an initial velocity of <math>\langle 3,0,0 \rangle m/s</math>. A constant net force of <math>\langle 4,0,0 \rangle N</math> acts on the cart for 2 seconds. What is the final momentum of the cart?
 
First find the initial momentum:
 
<math>\vec{p}_i = m\vec{v}</math>
 
<math>\vec{p}_i = 2\langle 3,0,0 \rangle = \langle 6,0,0 \rangle kg \cdot m/s</math>
 
Now find the impulse:
 
<math>\vec{J} = \vec{F}_{net}\Delta t</math>
 
<math>\vec{J} = \langle 4,0,0 \rangle(2) = \langle 8,0,0 \rangle N \cdot s</math>
 
Now use the impulse-momentum theorem:
 
<math>\vec{p}_f = \vec{p}_i + \vec{J}</math>
 
<math>\vec{p}_f = \langle 6,0,0 \rangle + \langle 8,0,0 \rangle</math>
 
<math>\vec{p}_f = \langle 14,0,0 \rangle kg \cdot m/s</math>
 
The final momentum of the cart is:
 
<math>\boxed{\langle 14,0,0 \rangle kg \cdot m/s}</math>
 
==Worked Example 2: Finding Average Force==
 
A 0.15 kg baseball is moving at 40 m/s before being hit. After contact with the bat, it moves in the opposite direction at 50 m/s. If the contact time is 0.01 seconds, what is the average force on the ball?
 
Let the original direction be positive. Then:
 
<math>v_i = 40 m/s</math>
 
<math>v_f = -50 m/s</math>
 
Find the change in momentum:
 
<math>\Delta p = m(v_f - v_i)</math>
 
<math>\Delta p = 0.15(-50 - 40)</math>
 
<math>\Delta p = 0.15(-90)</math>
 
<math>\Delta p = -13.5 kg \cdot m/s</math>
 
Now use:
 
<math>F_{avg} = \frac{\Delta p}{\Delta t}</math>
 
<math>F_{avg} = \frac{-13.5}{0.01}</math>
 
<math>F_{avg} = -1350 N</math>
 
The average force is:
 
<math>\boxed{1350N}</math>
 
The negative sign means the force acts opposite the ball's original direction.
 
==Common Mistakes==
 
* Forgetting that momentum and impulse are vectors.
* Forgetting to include direction when velocity changes.
* Confusing momentum with impulse.
* Using force instead of net force.
* Forgetting to multiply force by time.
* Thinking a bigger force always means a bigger impulse, even though time also matters.
* Forgetting that <math>N \cdot s</math> and <math>kg \cdot m/s</math> are equivalent units.
 
==Real-World Applications==
 
===Airbags===
 
Airbags reduce injury during car crashes by increasing the time it takes for a person's momentum to change. The person still needs to be brought to rest, so the change in momentum is the same. However, because the stopping time is longer, the average force on the person is smaller.
 
===Sports===
 
Impulse is important in sports such as baseball, tennis, soccer, and football. When a bat, racket, or foot stays in contact with a ball for a longer time, the impulse can increase. This changes the ball's momentum and can make it leave with a greater speed.
 
===Landing from a Jump===
 
When someone lands from a jump, bending the knees increases the stopping time. This lowers the average force on the person's legs. Landing with locked knees creates a shorter stopping time and a larger force.
 
==Computational Model==
 
In a computer simulation, momentum can be updated step by step using the momentum principle:
 
<pre>
p = p + Fnet*deltat
</pre>
 
This means that during each small time interval, the object's momentum changes because of the net force acting on it.
 
A simple GlowScript simulation for this topic could show a ball moving forward while a force acts on it for a short time. The simulation could display the ball's changing momentum after the impulse is applied.
 
==Practice Questions==
 
# A 3 kg object moves with velocity <math>\langle 2,0,0 \rangle m/s</math>. What is its momentum?
# A force of <math>\langle 10,0,0 \rangle N</math> acts on an object for 4 seconds. What impulse is delivered?
# A cart has initial momentum <math>\langle 5,0,0 \rangle kg \cdot m/s</math>. An impulse of <math>\langle -2,0,0 \rangle N \cdot s</math> acts on it. What is its final momentum?
 
==See Also==
 
* [[Linear Momentum]]
* [[Newton's Second Law: the Momentum Principle]]
* [[Conservation of Momentum]]
* [[Collisions]]
* [[Iterative Prediction]]


==References==
==References==
[http://www.physics.ohio-state.edu/~humanic/p111_lecture13]
[http://study.com/academy/lesson/momentum-and-impulse-definition-theorem-and-examples.html]
[http://www.physicsclassroom.com/class/momentum/Lesson-1/Real-World-Applications
]


[[Category:Momentum]]
* OpenStax. ''College Physics 2e'', Momentum and Impulse.
* The Physics Classroom. ''Momentum and Impulse Connection.''
* HyperPhysics. ''Impulse and Momentum.''
* Khan Academy. ''Impulse and Momentum.''

Latest revision as of 16:42, 27 April 2026

Edwyn Torres — Spring 2026

Impulse and Momentum

Introduction

Impulse and momentum are important ideas in physics because they help explain how forces change the motion of objects. Momentum describes how difficult it is to stop or change the motion of an object. Impulse describes how much a force changes an object's momentum over a period of time.

A large force acting for a short time can create the same impulse as a smaller force acting for a longer time. This idea is important in collisions, sports, car safety, and many other real-world situations.

Main Idea

Momentum is a vector quantity that depends on mass and velocity:

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

Impulse is also a vector quantity. It is the effect of a net force acting over a time interval:

[math]\displaystyle{ \vec{J} = \vec{F}_{net,avg}\Delta t }[/math]

The impulse-momentum theorem says that impulse equals the change in momentum:

[math]\displaystyle{ \vec{J} = \Delta \vec{p} }[/math]

This can also be written as:

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

This equation is useful because it lets us predict how an object's momentum changes when a force acts on it.

Key Equations

  • [math]\displaystyle{ \vec{p} = m\vec{v} }[/math]
  • [math]\displaystyle{ \vec{J} = \vec{F}_{net,avg}\Delta t }[/math]
  • [math]\displaystyle{ \vec{J} = \Delta \vec{p} }[/math]
  • [math]\displaystyle{ \Delta \vec{p} = \vec{p}_f - \vec{p}_i }[/math]
  • [math]\displaystyle{ \vec{p}_f = \vec{p}_i + \vec{F}_{net}\Delta t }[/math]

Where:

  • [math]\displaystyle{ \vec{p} }[/math] = momentum
  • [math]\displaystyle{ m }[/math] = mass
  • [math]\displaystyle{ \vec{v} }[/math] = velocity
  • [math]\displaystyle{ \vec{J} }[/math] = impulse
  • [math]\displaystyle{ \vec{F}_{net} }[/math] = net force
  • [math]\displaystyle{ \Delta t }[/math] = time interval

Units

The unit for momentum is:

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

The unit for impulse is:

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

These units are equivalent because:

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

So:

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

This makes sense because impulse equals change in momentum.

Conceptual Explanation

Impulse depends on two things: force and time. If the force is larger, the impulse is larger. If the force acts for a longer time, the impulse is also larger.

For example, when catching a baseball, a person usually moves their hand backward as the ball arrives. This increases the time over which the ball slows down. Since the ball still needs the same change in momentum to stop, increasing the stopping time decreases the average force on the hand.

This is why catching a ball with stiff hands hurts more than catching it while moving your hands backward.

Worked Example 1: Finding Final Momentum

A 2 kg cart is moving with an initial velocity of [math]\displaystyle{ \langle 3,0,0 \rangle m/s }[/math]. A constant net force of [math]\displaystyle{ \langle 4,0,0 \rangle N }[/math] acts on the cart for 2 seconds. What is the final momentum of the cart?

First find the initial momentum:

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

[math]\displaystyle{ \vec{p}_i = 2\langle 3,0,0 \rangle = \langle 6,0,0 \rangle kg \cdot m/s }[/math]

Now find the impulse:

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

[math]\displaystyle{ \vec{J} = \langle 4,0,0 \rangle(2) = \langle 8,0,0 \rangle N \cdot s }[/math]

Now use the impulse-momentum theorem:

[math]\displaystyle{ \vec{p}_f = \vec{p}_i + \vec{J} }[/math]

[math]\displaystyle{ \vec{p}_f = \langle 6,0,0 \rangle + \langle 8,0,0 \rangle }[/math]

[math]\displaystyle{ \vec{p}_f = \langle 14,0,0 \rangle kg \cdot m/s }[/math]

The final momentum of the cart is:

[math]\displaystyle{ \boxed{\langle 14,0,0 \rangle kg \cdot m/s} }[/math]

Worked Example 2: Finding Average Force

A 0.15 kg baseball is moving at 40 m/s before being hit. After contact with the bat, it moves in the opposite direction at 50 m/s. If the contact time is 0.01 seconds, what is the average force on the ball?

Let the original direction be positive. Then:

[math]\displaystyle{ v_i = 40 m/s }[/math]

[math]\displaystyle{ v_f = -50 m/s }[/math]

Find the change in momentum:

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

[math]\displaystyle{ \Delta p = 0.15(-50 - 40) }[/math]

[math]\displaystyle{ \Delta p = 0.15(-90) }[/math]

[math]\displaystyle{ \Delta p = -13.5 kg \cdot m/s }[/math]

Now use:

[math]\displaystyle{ F_{avg} = \frac{\Delta p}{\Delta t} }[/math]

[math]\displaystyle{ F_{avg} = \frac{-13.5}{0.01} }[/math]

[math]\displaystyle{ F_{avg} = -1350 N }[/math]

The average force is:

[math]\displaystyle{ \boxed{1350N} }[/math]

The negative sign means the force acts opposite the ball's original direction.

Common Mistakes

  • Forgetting that momentum and impulse are vectors.
  • Forgetting to include direction when velocity changes.
  • Confusing momentum with impulse.
  • Using force instead of net force.
  • Forgetting to multiply force by time.
  • Thinking a bigger force always means a bigger impulse, even though time also matters.
  • Forgetting that [math]\displaystyle{ N \cdot s }[/math] and [math]\displaystyle{ kg \cdot m/s }[/math] are equivalent units.

Real-World Applications

Airbags

Airbags reduce injury during car crashes by increasing the time it takes for a person's momentum to change. The person still needs to be brought to rest, so the change in momentum is the same. However, because the stopping time is longer, the average force on the person is smaller.

Sports

Impulse is important in sports such as baseball, tennis, soccer, and football. When a bat, racket, or foot stays in contact with a ball for a longer time, the impulse can increase. This changes the ball's momentum and can make it leave with a greater speed.

Landing from a Jump

When someone lands from a jump, bending the knees increases the stopping time. This lowers the average force on the person's legs. Landing with locked knees creates a shorter stopping time and a larger force.

Computational Model

In a computer simulation, momentum can be updated step by step using the momentum principle: 

p = p + Fnet*deltat

This means that during each small time interval, the object's momentum changes because of the net force acting on it.

A simple GlowScript simulation for this topic could show a ball moving forward while a force acts on it for a short time. The simulation could display the ball's changing momentum after the impulse is applied.

Practice Questions

  1. A 3 kg object moves with velocity [math]\displaystyle{ \langle 2,0,0 \rangle m/s }[/math]. What is its momentum?
  2. A force of [math]\displaystyle{ \langle 10,0,0 \rangle N }[/math] acts on an object for 4 seconds. What impulse is delivered?
  3. A cart has initial momentum [math]\displaystyle{ \langle 5,0,0 \rangle kg \cdot m/s }[/math]. An impulse of [math]\displaystyle{ \langle -2,0,0 \rangle N \cdot s }[/math] acts on it. What is its final momentum?

See Also

References

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