Difference between revisions of "Forest UCM MnAM ElasticCol"

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:<math>\vec{v}_1 \cdot \vec{v}_1 =  \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right ) \cdot  \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right )</math>
 
:<math>\vec{v}_1 \cdot \vec{v}_1 =  \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right ) \cdot  \left ( \vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime}\right )</math>
 
:<math>v_1^2 = \left(v_1^{\prime}\right)^2 + \left(v_2^{\prime}\right)^2 + 2  \vec{v}_1^{\;\prime} \cdot  \vec{v}_2^{\;\prime}</math>  
 
:<math>v_1^2 = \left(v_1^{\prime}\right)^2 + \left(v_2^{\prime}\right)^2 + 2  \vec{v}_1^{\;\prime} \cdot  \vec{v}_2^{\;\prime}</math>  
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In order for both the conservation of momentum and energy properties for eleastic collisions to hold
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:<math>\vec{v}_1^{\;\prime} \cdot  \vec{v}_2^{\;\prime} = 0 </math>
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or the balls make an angle of 90 degrees with repect to eachother
  
  
 
[[Forest_UCM_MnAM#Elastic_Collision_of_2_bodies]]
 
[[Forest_UCM_MnAM#Elastic_Collision_of_2_bodies]]

Revision as of 12:20, 12 September 2014

An Elastic collision conserves both Momentum and Energy

Pinitial=Pfinal
AND
Einitial=Efinal

Example: problem 3.5

Consider an elastic collision of two equal balls of mass m where one ball has an initial velocity v1 and the remaining ball has zero initial velocity.

Determine the angle between the two balls after the collision.


Conservation of momentum
m1v1+m2v2=m1v1+m2v2
m1v1=m1v1+m2v2: ball 2 has zero velocity
v1=v1+v2: balls have equal masses


Conservation of energy
m1v21+m2v22=m1(v1)2+m2(v2)2
v21=(v1)2+(v2)2

If I look at the dot product of the conservation of momentum equation

v1=v1+v2: balls have equal masses
v1v1=(v1+v2)(v1+v2)
v21=(v1)2+(v2)2+2v1v2


In order for both the conservation of momentum and energy properties for eleastic collisions to hold

v1v2=0

or the balls make an angle of 90 degrees with repect to eachother


Forest_UCM_MnAM#Elastic_Collision_of_2_bodies