Impulse and Momentum

Impulse Momentum

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

Impulse Momentum Theorem

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).

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]\displaystyle{ {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]\displaystyle{ {F} = {\frac{d\vec{p}}{dt}} }[/math], which shows that whenever momentum changes with time, there is some force acting on the body.

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.

Real World Applications

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: [1]

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. [2]

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.

A Mathematical Model

Impulse can mathematically be defined as the force on a body multiplied by the duration of that force. [math]\displaystyle{ {\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]\displaystyle{ {J} = {d\vec{p}} = \vec{F}_{net}{dt} }[/math]

A Computational Model

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.

 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)

Examples

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

Simple

Q: What is the impulse imparted by a rocket that exerts 4.8 N for 1.63 seconds?

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

Middling

This physics teacher works out a slightly more involved impulse collision analysis on a real car crash from a safety video. Follow along here: [3]

Difficult

For you engineers, here is a more advanced impulse problem using calculus to solve a mechanics problem: [4]

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.

History

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

See also

A great video that explains this concept with visual examples: [5]

[6]

Further reading

RANKINE, William John Macquorn, and Edward Fisher BAMBER. A Mechanical Text Book. 1873. Print.

External links

[http://www.physicsclassroom.com/class/momentum/Lesson-1/Momentum-and-Impulse-Connection ]

References

[7] [8] [http://www.physicsclassroom.com/class/momentum/Lesson-1/Real-World-Applications ]