ch6_arccomrm

Chapter 6 - Momentum and Collisions

Section One - Momentum and Impulse Momentum is a vector quantity described by the product of an object's mass times its velocity: math \displaystyle \vec{p}=m\vec{v} math If an object's momentum is known, its kinetic energy can be found as follows: math \displaystyle KE=\frac{p^2}{2m} math The change in the momentum of an object is equal to the impulse delivered to the object. The impulse is equal to the constant net external force acting on the object times the time over which the force acts: math \displaystyle \Delta\vec{p}=\vec{F}\Delta t math Any force acting on an object will cause an impulse, including frictional and gravitational forces In one dimension the slope of a graph of the momentum of an object vs. time is the net external force acitng on the object. The area under a graph of the net external force acting on an object vs. time is the total change in momentum of that object. Momentum and impulse are measured in: math \displaystyle kg\frac{m}{s} math -there is no special name for that unit

 Section Two - Conservation of Momentum The law of conservation of momentum is the total momentum of all objects interacting with one another remains constant regardless of the nature of the force between the objects:

math \displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f math which is: math \displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f math

  This relationship is true for all interactions between isolated objects. The change in momentum of the first object is equal to and opposite the change in momentum of the second object.

//An Explanation of the Conservation of momentum:// -To get to the conservation of momentum problem, you must consider Newton's Third law: Newton's Third law leads to conservation of momentum. It tells us that the change in momentum of the first object is equal to and opposite the change in momentum of the second object. -The force on m1 is equal to and opposite the force on m2 (f1= - f2), the impulse on m1 is equal to and opposite the impulse on m2; therefore the change in momentum of m1 is equal to and opposite the change in momentum m2.



Image from: http://webpages.uah.edu/~wilderd/momentum.jpg

=

 * //Elastic collision//**: A collision in which the total momentum and the total kinetic energy remain constant; the two objects return to their original shapes with no change in total kinetic energy or momentum; no energy is lost ======

\displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f math \displaystyle \Sigma KE_i>\Sigma KE_f math
 * math
 * math

=
//**Perfectly Inelastic collision**//: A collision in which two objects stick together and move with a common velocity after colliding; the final mass is equal to the combined masses of the two objects.======

\displaystyle m_1\vec{v}_{1i}+m_2\vec{v}_{2i}=m_1+\vec{v}_{1f}+m_2\vec{v}_{2f} math \displaystyle \frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2>\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2 math
 * math
 * math

=

 * Inelastic collsion**: the total kinetic energy does not remain the constant when the objects collide and stick together; the two objects in the collision are deformed and lose some kinetic energy. The decrease in kinetic energy can be calculated using the formula for kinetic energy from Chapter 5.======



General Review __Major Equations:__
\displaystyle \vec{p}=m\vec{v} math \displaystyle KE={p^2}{2m} math \displaystyle \Delta\vec{p}=\vec{F}\Delta t math \displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f math \displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f math \displaystyle \Delta\vec{p}_1=-\Delta\vec{p}_2 math \displaystyle \Sigma\vec{p}_1=\Sigma\vec{p}_f math \displaystyle \Sigma KE_i=\Sigma KE_f math \displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f math \displaystyle \frac{1}{2}m_1v_1^2_i+\frac{1}{2}m_2v_2^2_i=\frac{1}{2}m_1v_1^2_f+\frac{1}{2}m_2v_2^2_f math \displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f math \displaystyle \Sigma KE_i>\Sigma KE_f math \displaystyle m_1\vec{v}_{1i}+m_2\vec{v}_{2i}=m_1+\vec{v}_{1f}+m_2\vec{v}_{2f} math \displaystyle \frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2>\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2 math
 * Momentum
 * math
 * Kenetic Energy (Given Momentum)
 * math
 * Impulse
 * math
 * Conservation of Momentum (General)
 * math
 * Conservation of Momentum (Two Objects)
 * math
 * Change in Momentum
 * math
 * Elastic Collisions (General)
 * math
 * math
 * Elastic Collisions (Two Objects)
 * math
 * math
 * Inelastic Collisions (General)
 * math
 * math
 * Inelastic Collisions (Two Objects)
 * math
 * math

Information from Holt Physics textbook