Suggestions
Use up and down arrows to review and enter to select.Please wait while we process your payment
If you don't see it, please check your spam folder. Sometimes it can end up there.
If you don't see it, please check your spam folder. Sometimes it can end up there.
Please wait while we process your payment
By signing up you agree to our terms and privacy policy.
Don’t have an account? Subscribe now
Create Your Account
Sign up for your FREE 7-day trial
Already have an account? Log in
Your Email
Choose Your Plan
Individual
Group Discount
Save over 50% with a SparkNotes PLUS Annual Plan!
Purchasing SparkNotes PLUS for a group?
Get Annual Plans at a discount when you buy 2 or more!
Price
$24.99 $18.74 /subscription + tax
Subtotal $37.48 + tax
Save 25% on 2-49 accounts
Save 30% on 50-99 accounts
Want 100 or more? Contact us for a customized plan.
Your Plan
Payment Details
Payment Summary
SparkNotes Plus
You'll be billed after your free trial ends.
7-Day Free Trial
Not Applicable
Renews October 11, 2023 October 4, 2023
Discounts (applied to next billing)
DUE NOW
US $0.00
SNPLUSROCKS20 | 20% Discount
This is not a valid promo code.
Discount Code (one code per order)
SparkNotes PLUS Annual Plan - Group Discount
Qty: 00
SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.
Choose Your Plan
For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!
You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.
Members will be prompted to log in or create an account to redeem their group membership.
Thanks for creating a SparkNotes account! Continue to start your free trial.
Please wait while we process your payment
Your PLUS subscription has expired
Please wait while we process your payment
Please wait while we process your payment
What happens when a group of particles are all interacting? Qualitatively speaking, each exerts equal and opposite impulses on the other, and though the individual momentum of any given particle might change, the total momentum of the system remains constant. This phenomenon of momentum constancy describes the conservation of linear momentum in a nutshell; in this section we shall prove the existence of the conservation of energy by using what we already know about momentum and systems of particles.
Just as we first defined kinetic energy for a single particle, and then examined the energy of a system, so shall we now turn to the linear momentum of a system of particles. Suppose we have a system of N particles, with masses m1, m2, , mn. Assuming no mass enters or leaves the system, we define the total momentum of the system as the vector sum of the individual momentum of the particles:
P | = | p1 + p2 + ... + pn | |
= | m1v1 + m2v2 + ... + mnvn |
P = Mvcm |
![]() ![]() |
From our last equation we will consider now the special case in which Fext = 0. That is, no external forces act upon an isolated system of
particles. Such a situation implies that the rate of change of the total
momentum of a system does not change, meaning this quantity is constant, and
proving the principle of the conservation of linear momentum:
When there is no net external force acting on a system of particles the total momentum of the system is conserved.
It's that simple. No matter the nature of the interactions that go on within a given system, its total momentum will remain the same. To see exactly how this concept works we shall consider an example.
Let's consider a cannon firing a cannonball. Initially, both the cannon and the ball are at rest. Because the cannon, the ball, and the explosive are all within the same system of particles, we can thus state that the total momentum of the system is zero. What happens when the cannon is fired? Clearly the cannonball shoots out with considerable velocity, and thus momentum. Because there are no net external forces acting on the system, this momentum must be compensated for by a momentum in the opposite direction as the velocity of the ball. Thus the cannon itself is given a velocity backwards, and total momentum is conserved. This conceptual example accounts for the "kick" associated with firearms. Any time a gun, a cannon, or an artillery piece releases a projectile, it must itself move in the direction opposite the projectile. The heavier the firearm, the slower it moves. This is a simple example of the conservation of linear momentum.
By both examining the center of mass of a system of particles, and developing the conservation of linear momentum we can account for a great deal of motion in a system of particles. We now know how to calculate both the motion of the system as a whole, based on external forces applied to the system, and the activity of the particles within the system, based on momentum conservation within the system. This topic, dealing with momentum, is as important as the last one, dealing with energy. Both concepts are universally applied: while Newton's Laws apply only to mechanics, conservation of momentum and energy are used in relativistic and quantum calculations as well.
Please wait while we process your payment