A Collison

• A hockey puck with mass 4 kg makes a head on collision with a smaller puck with mass 2 kg. The larger puck was originally travelling east at 3 m/sec. The smaller puck was originally travelling west at 4 m/sec.

• How fast are they moving after the collision. (Assume a perfectly elastic collision all in a straight line, that is one dimensional.)

Fundamentals, Using Math• Before collision: mA=4 vA=3 mB=2 vB=−4

• After collision: mA=4 vA=? mB=2 vB = ?

• Momentum In (mv) = Momentum Out (mv)• 4 3 + 2 (−4) = 4 v∙ ∙ A + 2vB

• KE In (½mv2) = KE Out (½mv2)• ½ 4 3∙ ∙ 2 + ½ 2 (−4)∙ ∙ 2 = ½ 4v∙ A

2 + ½ 2v∙ B2

• 18 + 16 = 2 vA2 + vB

2

• 4vA + 2vB = 4 (momentum)

• 2vA2 + vB

2 = 34 (energy)

Solving Simultaneously

• 4 vA + 2vB = 4 (momentum)

• vB = 2 − 2vA

• 2vA2 + vB

2 = 34 (energy)

• 2vA2 + (2 − 2vA)2 = 34

• 2vA2 + 4 − 2 4v∙ A + 4vA

2 = 34

• 6vA2 − 8vA − 30 = 0

• 2 (3v∙ A2 − 4vA − 15) = 0

• 2 (3v∙ A + 5) (v∙ A − 3) = 0

Solutions

• 2 (3v∙ A + 5) (v∙ A − 3) = 0• One of the factors must be zero• 3vA + 5 = 0 vA = − 5/3

• vB = 2−2vA = 16/3

• Or vA − 3 = 0 vA = 3

• vB = 2−2vA = − 4

CheatingReading the Book

• Galileo was challenged. “If the earth is moving so fast around the sun, why don’t we feel it.”

• He responded by describing a goldfish in its bowl in the captain’s cabin on a ship.

• Does the goldfish swim faster in the direction of the ship’s motion?

• Galilean Relativity • The Laws of Physics are the same if we add the

same amount to all velocities.

If vB1 is Zero• mAvA1 = mAvA2 + mBvB2 (momentum)

• ½mAvA1² = ½mAvA2² + ½ mBvB2² (energy)

• mAvA1² − mAvA2² = mBvB2²

• mAvA1 − mAvA2 = mBvB2

• (vA1² − vA2²)/(vA1 − vA2) = vB2

• vA2 + vA1 = vB2 *** (Elastic Collisions)• According to Galileo, we can add the same quantity to all

velocities, and get a valid equation.• If we start with vB1 not equal to zero, we can add −vB1 to all

quantities. Then equation *** is true • vA2−vB1 + vA1−vB1 = vB2 −vB1

• vA1 − vB1 = vB2 − vA2

The Short Way (Cheating)

• vA1 − vB1 = vB2 − vA2

• 3 − (−4) = vB2 − vA2

• vB2 − vA2 = 7

• 4vA2 + 2vB2 = 4 (momentum)

• vB2 = vA2 + 7

• 4vA2 + 2(vA2 + 7) = 4

• 6vA2 = −10

• vA2 = −5/3

• vB2 = 16/3