Momentum Part 3 By: Heather Britton

Elastic Collisions Elastic collisions are a special type of collisions that do not often occur in everyday experience Elastic collision - one in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision Not only is momentum conserved, but kinetic energy is also conserved Elastic collisions are a special type of collisions that do not often occur in everyday experience Elastic collision - one in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision Not only is momentum conserved, but kinetic energy is also conserved

Elastic Collisions p 1o + p 2o = p 1 + p 2 (mv) 1o + (mv) 2o = (mv) 1 + (mv) 2 KE 1o + KE 2o = KE 1 + KE 2 (1/2)(mv 2 ) 1o + (1/2)(mv 2 ) 2o = (1/2)(mv 2 ) 1 +(1/2)(mv 2 ) 2 p 1o + p 2o = p 1 + p 2 (mv) 1o + (mv) 2o = (mv) 1 + (mv) 2 KE 1o + KE 2o = KE 1 + KE 2 (1/2)(mv 2 ) 1o + (1/2)(mv 2 ) 2o = (1/2)(mv 2 ) 1 +(1/2)(mv 2 ) 2

Elastic Collisions Solving problems concerning collisions will involve using simultaneous equations With elastic collisions we can derive an equation to represent the velocities before and after the event We will use p o = p and KE o = KE Solving problems concerning collisions will involve using simultaneous equations With elastic collisions we can derive an equation to represent the velocities before and after the event We will use p o = p and KE o = KE

Elastic Collisions (1/2)(mv 2 ) 1o + (1/2)(mv 2 ) 2o = (1/2)(mv 2 ) 1 +(1/2)(mv 2 ) 2 Canceling out the (1/2) gives (mv 2 ) 10 + (mv 2 ) 2o = (mv 2 ) 1 + (mv 2 ) 2 Rearranging gives (mv 2 ) 1o - (mv 2 ) 1 = (mv 2 ) 2 - (mv 2 ) 2o (1/2)(mv 2 ) 1o + (1/2)(mv 2 ) 2o = (1/2)(mv 2 ) 1 +(1/2)(mv 2 ) 2 Canceling out the (1/2) gives (mv 2 ) 10 + (mv 2 ) 2o = (mv 2 ) 1 + (mv 2 ) 2 Rearranging gives (mv 2 ) 1o - (mv 2 ) 1 = (mv 2 ) 2 - (mv 2 ) 2o

Elastic Collisions Factoring out the m gives m 1 (v 1o 2 - v 1 2 ) = m 2 (v v 2o 2 ) Factoring the squares gives m 1 (v 1o - v 1 )(v 1o + v 1 ) = m 2 (v 2 - v 2o )(v 2 + v 2o ) Factoring out the m gives m 1 (v 1o 2 - v 1 2 ) = m 2 (v v 2o 2 ) Factoring the squares gives m 1 (v 1o - v 1 )(v 1o + v 1 ) = m 2 (v 2 - v 2o )(v 2 + v 2o )

Elastic Collisions Now lets play with the conservation of momentum equation (mv) 1o + (mv) 2o = (mv) 1 + (mv) 2 Rearranging gives (mv) 1o - (mv) 1 = (mv) 2 - (mv) 2o Now lets play with the conservation of momentum equation (mv) 1o + (mv) 2o = (mv) 1 + (mv) 2 Rearranging gives (mv) 1o - (mv) 1 = (mv) 2 - (mv) 2o

Elastic Collisions Factoring out the m gives m 1 (v 1o - v 1 ) = m 2 (v 2 - v 2o ) Now divide the conservation of energy equation by the conservation of momentum equation to get [m 1 (v 1o - v 1 )(v 1o + v 1 ) = m 2 (v 2 - v 2o )(v 2 + v 2o )] / [m 1 (v 1o - v 1 ) = m 2 (v 2 - v 2o )] Factoring out the m gives m 1 (v 1o - v 1 ) = m 2 (v 2 - v 2o ) Now divide the conservation of energy equation by the conservation of momentum equation to get [m 1 (v 1o - v 1 )(v 1o + v 1 ) = m 2 (v 2 - v 2o )(v 2 + v 2o )] / [m 1 (v 1o - v 1 ) = m 2 (v 2 - v 2o )]

Elastic Collisions Canceling out and simplifying gives v 1o + v 1 = v 2 + v 2o Rearrange again to arrive at our destination v 1o - v 2o = -(v 1 - v 2 ) This is only valid for an elastic collision Canceling out and simplifying gives v 1o + v 1 = v 2 + v 2o Rearrange again to arrive at our destination v 1o - v 2o = -(v 1 - v 2 ) This is only valid for an elastic collision

Example 6 A kg ice puck, moving east with a speed of 3.7 m/s, has a head-on collision with a kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?