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- Date Submitted: 08/18/2011 03:43 PM
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Conservation of Energy and Linear Momentum in Collisions
← Total kinetic energy before the collision and the total kinetic energy after the collision is the same or energy is conserved ( elastic collision
½ m1 v1i 2 + ½ m2 v2i 2 = ½ m1 v1f 2 + ½ m2 v2f 2 …………[1]
← Collisions in which kinetic energy is not conserved are said to be inelastic collisions.
← From conservation of momentum
m1 v1i + m2 v2i = m1 v1f + m2 v2f …………[2]
2 equations, solve for two unknowns.
Rewrite the momentum equations as
m1 ( v1i - v1f ) = m2 ( v2f - v2i ) …………[3]
Rewrite the kinetic energy equations as
m1 ( v1i 2 - v1f 2 ) = m2 ( v2f 2 - v2i 2 ) ………..[4]
[Noting that (a – b)(a + b) = a2 – b2], we write equation [4] as
m1 ( v1i - v1f )( v1i - v1f ) = m2 ( v2f - v2i )( v2f + v2i ) ……….[5]
Divide equation [5] by equation [3], we get
v1i + v1f = v2f + v2i ……….[6]
Rewrite this equation as
v1i - v2i = v2f - v1f = - ( v1f - v2f ) ……….[7]
Conservation of momentum ( v2i = 0)
m1 v1i = m1 v1f + m2 v2f ……….[8]
Conservation of kinetic energy
½ m1 v1i 2 = ½ m1 v1f 2 + ½ m2 v2f 2 ………..[9]
From equation [6]
v2f = v1i + v1f
v1f = v2f - v1i ……….[10]
Insert equation [10] into equation [8]
m1 v1i = m1 ( v2f - v1i ) + m2 v2f
m1 v1i = m1 v2f - m1 v1i + m2 v2f
m1 v1i + m1 v1i = m1 v2f + m2 v2f
2 m1 v1i = ( m1 + m2 ) v2f
[pic] ……….[11]
and inserting v2f = v1i + v1f into the equation for conservation of momentum [8]
m1 v1i = m1 v1f + m2 ( v1f + v1i )
= m1...
Conservation of Energy and Linear Momentum in Collisions
← Total kinetic energy before the collision and the total kinetic energy after the collision is the same or energy is conserved ( elastic collision
½ m1 v1i 2 + ½ m2 v2i 2 = ½ m1 v1f 2 + ½ m2 v2f 2 …………[1]
← Collisions in which kinetic energy is not conserved are said to be inelastic collisions.
← From conservation of momentum
m1 v1i + m2 v2i = m1 v1f + m2 v2f …………[2]
2 equations, solve for two unknowns.
Rewrite the momentum equations as
m1 ( v1i - v1f ) = m2 ( v2f - v2i ) …………[3]
Rewrite the kinetic energy equations as
m1 ( v1i 2 - v1f 2 ) = m2 ( v2f 2 - v2i 2 ) ………..[4]
[Noting that (a – b)(a + b) = a2 – b2], we write equation [4] as
m1 ( v1i - v1f )( v1i - v1f ) = m2 ( v2f - v2i )( v2f + v2i ) ……….[5]
Divide equation [5] by equation [3], we get
v1i + v1f = v2f + v2i ……….[6]
Rewrite this equation as
v1i - v2i = v2f - v1f = - ( v1f - v2f ) ……….[7]
Conservation of momentum ( v2i = 0)
m1 v1i = m1 v1f + m2 v2f ……….[8]
Conservation of kinetic energy
½ m1 v1i 2 = ½ m1 v1f 2 + ½ m2 v2f 2 ………..[9]
From equation [6]
v2f = v1i + v1f
v1f = v2f - v1i ……….[10]
Insert equation [10] into equation [8]
m1 v1i = m1 ( v2f - v1i ) + m2 v2f
m1 v1i = m1 v2f - m1 v1i + m2 v2f
m1 v1i + m1 v1i = m1 v2f + m2 v2f
2 m1 v1i = ( m1 + m2 ) v2f
[pic] ……….[11]
and inserting v2f = v1i + v1f into the equation for conservation of momentum [8]
m1 v1i = m1 v1f + m2 ( v1f + v1i )
= m1...