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**Chapter 9B - Conservation of Momentum**

A PowerPoint Presentation by Paul E. Tippens, Emeritus Professor Southern Polytechnic State University © 2007

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**Momentum is conserved in this rocket launch**

Momentum is conserved in this rocket launch. The velocity of the rocket and its payload is determined by the mass and velocity of the expelled gases. Photo: NASA NASA

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**Objectives: After completing this module, you should be able to:**

State the law of conservation of momentum and apply it to the solution of problems. Distinguish by definition and example between elastic and inelastic collisions. Predict the velocities of two colliding bodies when given the coefficients of restitution, masses, and initial velocities.

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**A Collision of Two Masses**

When two masses m1 and m2 collide, we will use the symbol u to describe velocities before collision. m1 u1 m2 u2 Before The symbol v will describe velocities after collision. After m1 v1 m2 v2

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**A Collision of Two Blocks**

m1 u1 m2 u2 Before Collision m1 B m2 “u”= Before “v” = After m2 v2 m1 v1 After

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**Conservation of Energy**

m1 m2 u1 u2 The kinetic energy before colliding is equal to the kinetic energy after colliding plus the energy lost in the collision.

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Example 1. A 2-kg mass moving at 4 m/s collides with a 1-kg mass initially at rest. After the collision, the 2-kg mass moves at 1 m/s and the 1-kg mass moves at 3 m/s. What energy was lost in the collision? It’s important to draw and label a sketch with appropriate symbols and given information. m2 u2 = 0 m1 u1 = 4 m/s m1 = 2 kg m1 = 1 kg BEFORE m2 v2 = 2 m/s m1 v1 = 1 m/s m1 = 2 kg m1 = 1 kg AFTER

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**Energy Conservation: K(Before) = K(After) + Loss**

Example 1 (Continued). What energy was lost in the collision? Energy is conserved. m2 u2 = 0 m1 u1 = 4 m/s m1 = 2 kg m1 = 1 kg v2 = 2 m/s v1 = 1 m/s BEFORE: AFTER Energy Conservation: K(Before) = K(After) + Loss Loss = 16 J – 3 J Energy Loss = 15 J

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**mBvB - mBuB = -(mAvA - mAuA)**

Impulse and Momentum A B uA uB vA vB Impulse = Dp -FA Dt FB Dt FDt = mvf– mvo Opposite but Equal F Dt FBDt = -FADt mBvB - mBuB = -(mAvA - mAuA) Simplifying: mAvA + mBvB = mAuA + mBuB

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**Conservation of Momentum**

The total momentum AFTER a collision is equal to the total momentum BEFORE. mAvA + mBvB = mAuA + mBuB A B uA uB vA vB -FADt FB Dt Recall that the total energy is also conserved: Kinetic Energy: K = ½mv2 KA0 + KB0 = KAf + KBf + Loss

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**mAvA + mBvB = mAuA + mBuB mBvB mA mAvA = - mBvB vA = -**

Example 2: A 2-kg block A and a 1-kg block B are pushed together against a spring and tied with a cord. When the cord breaks, the 1-kg block moves to the right at 8 m/s. What is the velocity of the 2 kg block? A B The initial velocities are zero, so that the total momentum before release is zero. mAvA + mBvB = mAuA + mBuB vA = - mBvB mA mAvA = - mBvB

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**Example 2 (Continued) vA2 mAvA+ mBvB = mAuA + mBuB mBvB mA**

mAvA = - mBvB vA = - mBvB mA A B 2 kg 1 kg 8 m/s vA2 vA = - (1 kg)(8 m/s) (2 kg) vA = - 4 m/s

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**Example 2 (Cont.): Ignoring friction, how much energy was released by the spring?**

2 kg 1 kg A B 8 m/s 4 m/s Cons. of E: ½kx2 = ½ mAvA + ½mBvB2 2 ½kx2 = ½(2 kg)(4 m/s)2 + ½(1 kg)(8 m/s)2 ½kx2 = 48 J ½kx2 = 16 J + 32 J = 48 J

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Elastic or Inelastic? An elastic collision loses no energy. The deform-ation on collision is fully restored. In an inelastic collision, energy is lost and the deformation may be permanent. (Click it.)

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**Completely Inelastic Collisions**

Collisions where two objects stick together and have a common velocity after impact. Before After

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Example 3: A 60-kg football player stands on a frictionless lake of ice. He catches a 2-kg football and then moves at 40 cm/s. What was the initial velocity of the football? A Given: uB= 0; mA= 2 kg; mB= 60 kg; vA= vB= vC vC = 0.4 m/s B Momentum: mAvA + mBvB = mAuA + mBuB Inelastic collision: (mA + mB)vC = mAuA (2 kg + 60 kg)(0.4 m/s) = (2 kg)uA uA= 12.4 m/s

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**Example 3 (Cont.): How much energy was lost in catching the football?**

½(2 kg)(12.4 m/s)2 = ½(62 kg)(0.4 m/s)2 + Loss 154 J = 4.96 J + Loss Loss = 149 J 97% of the energy is lost in the collision!!

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**General: Completely Inelastic**

Collisions where two objects stick together and have a common velocity vC after impact. Conservation of Momentum: Conservation of Energy:

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**Common speed after colliding: 2.4 m/s.**

Example 4. An 87-kg skater B collides with a 22-kg skater A initially at rest on ice. They move together after the collision at 2.4 m/s. Find the velocity of the skater B before the collision. A B uB = ? uA = 0 Common speed after colliding: 2.4 m/s. 22 kg 87 kg vB= vA = vC = 2.4 m/s (87 kg)uB = (87 kg + 22 kg)(2.4 m/s) (87 kg)uB =262 kg m/s uB = 3.01 m/s

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Example 5: A 50 g bullet strikes a 1-kg block, passes all the way through, then lodges into the 2 kg block. Afterward, the 1 kg block moves at 1 m/s and the 2 kg block moves at 2 m/s. What was the entrance velocity of the bullet? 1 kg 2 kg uA= ? 2 kg 1 kg 1 m/s 2 m/s

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**mAuA + mBuB + mCuC = mBvB + (mA+mC) vAC**

50 g Find entrance velocity of bullet: mA= 0.05 kg; uA= ? 2 kg 1 kg 1 m/s 2 m/s Momentum After = Momentum Before mAuA + mBuB + mCuC = mBvB + (mA+mC) vAC (0.05 kg)uA =(1 kg)(1 m/s)+(2.05 kg)(2 m/s) (0.05 kg) uA =(5.1 kg m/s) uA= 102 m/s

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**Completely Elastic Collisions**

Collisions where two objects collide in such a way that zero energy is lost in the process. APPROXIMATIONS!

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**Velocity in Elastic Collisions**

uB uA 1. Zero energy lost. A B 2. Masses do not change. vA vB A B 3. Momentum conserved. Equal but opposite impulses (F Dt) means that: (Relative Dv After) = - (Relative Dv Before) vA - vB = - (uA - uB) For elastic collisions:

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**From conservation of energy (relative v):**

Example 6: A 2-kg ball moving to the right at 1 m/s strikes a 4-kg ball moving left at 3 m/s. What are the velocities after impact, assuming complete elasticity? A B 3 m/s 1 m/s vA vB 1 kg 2 kg vA - vB = - (uA - uB) vA - vB = uB - uA vA - vB = (-3 m/s) - (1 m/s) From conservation of energy (relative v): vA - vB = - 4 m/s

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**Example 6 (Continued) mAvA + mBvB = mAuA + mBuB vA + 2vB = -5 m/s**

1 kg 2 kg Energy: vA - vB = - 4 m/s Momentum also conserved: mAvA + mBvB = mAuA + mBuB (1 kg)vA+(2 kg)vB=(1 kg)(1 m/s)+(2 kg)(-3 m/s) vA + 2vB = -5 m/s Two independent equations to solve: vA - vB = - 4 m/s

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**Example 6 (Continued) vA + 2vB = -5 m/s vA - vB = - 4 m/s vA**

1 kg 2 kg vA + 2vB = -5 m/s vA - vB = - 4 m/s Subtract: 0 + 3vB2 = - 1 m/s vB = m/s vA2 - ( m/s) = - 4 m/s Substitution: vA= m/s vA - vB = - 4 m/s

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**(0.150 kg)vA+ (2 kg)(40 m/s) = (0.150 kg)(715 m/s)**

Example 7. A kg bullet is fired at 715 m/s into a 2-kg wooden block at rest. The velocity of block afterward is 40 m/s. The bullet passes through the block and emerges with what velocity? B A uB = 0 (0.150 kg)vA+ (2 kg)(40 m/s) = (0.150 kg)(715 m/s) 0.150vA+ (80 m/s) = (107 m/s) 0.150vA = 27.2 m/s) vA = 181 m/s

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**Example 8a: Inelastic collision: Find vC.**

B 5 kg 7.5 kg uB=0 2 m/s After hit: vB= vA= vC (5 kg)(2 m/s) = (5 kg kg)vC A B Common vC after vC 12.5 vC =10 m/s vC = m/s In an completely inelastic collision, the two balls stick together and move as one after colliding.

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**Example 8. (b) Elastic collision: Find vA2 and vB2**

5 kg 7.5 kg vB1=0 2 m/s Conservation of Momentum: (5 kg)(2 m/s) = (5 kg)vA2 + (7.5 kg) vB A B vA vB 5 vA vB = 10 m/s For Elastic Collisions: Continued . . .

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**Example 8b (Cont). Elastic collision: Find vA & vB**

Solve simultaneously: A B 5 kg 7.5 kg vB =0 2 m/s vA vB x (-5) 5 vA v B = 10 m/s 5 vA vB = 10 m/s -5 vA vB = +10 m/s vA m/s = -2 m/s vA = m/s 12.5 vB = 20 m/s vB = 1.60 m/s

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**General: Completely Elastic**

Collisions where zero energy is lost during a collision (an ideal case). Conservation of Momentum: Conservation of Energy:

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**The ballistic pendulum!**

Example 9: A 50 g bullet lodges into a 2-kg block of clay hung by a string. The bullet and clay rise together to a height of 12 cm. What was the velocity of the 50-g mass just before entering? uA B A B A 12 cm The ballistic pendulum!

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**Example (Continued): mAuA+0= (mA+mB)vC vC = 2gh**

Collision and Momentum: B A 12 cm 50 g uA 2.05 kg 2 kg mAuA+0= (mA+mB)vC (0.05 kg)uA = (2.05 kg)vC To find vA we need vC . After collision, energy is conserved for masses. vC = 2gh

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**Example (Continued): mAuA+0= (mA+mB)vC uA = 62.9 m/s**

vC = 2gh = 2(9.8)(0.12) B A 12 cm 50 g uA 2.05 kg 2 kg After Collision: vC = 1.53 m/s Momentum Also Conserved: mAuA+0= (mA+mB)vC (0.05 kg)uA = (2.05 kg)(1.53 m/s) uA = 62.9 m/s

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**Conservation of Momentum:**

Summary of Formulas: Conservation of Momentum: Conservation of Energy: For elastic only:

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**CONCLUSION: Chapter 9B Conservation of Momentum**

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