UCSD Physics 10

Momentum & ImpulseMomentum & Impulse

Winter 2006 2

UCSD Physics 10

Momentum, pMomentum, p

The linear momentum p of an object is the product of the object’s mass m and velocity v:

Linear momentum is a vector quantity that points in the same direction as the velocity.

SI Unit of Linear Momentum:

kilogram · meter/second = (kg · m/s)

Winter 2006 3

UCSD Physics 10

Impulse, JImpulse, J

The impulse J of a force is the product of the average force and the time interval t during which the force acts:

Impulse is a vector quantity and has the same direction as the average force.

SI Unit of Impulse: newton · second = (N · s)

Winter 2006 4

UCSD Physics 10

IMPULSE–MOMENTUM THEOREMIMPULSE–MOMENTUM THEOREM

When a net force acts on an object, the impulse of the net force is equal to the change in momentum of the object:

IMPUSE = CHANGE IN MOMENTUM

Winter 2006 5

UCSD Physics 10

Derivation of the Impulse-Momentum Derivation of the Impulse-Momentum theoremtheorem

• Start by applying Newton’s 3Start by applying Newton’s 3rdrd Law. Law.• F1 + F2 = 0; Equal and opposite forces!

• Expand the Equation!Expand the Equation!• m1a1 + m2 a2 = 0; but a = Δv/t

• m1Δv1/t + m2 Δv2/t = 0; cancel out time.

• Therefore, mTherefore, m11ΔΔvv11 + m + m2 2 ΔΔvv22 = 0 = 0

• Conservation of Linear Momentum.Conservation of Linear Momentum.

Winter 2006 6

UCSD Physics 10

Linear MomentumLinear Momentum

• Often misused word, though most have the right ideaOften misused word, though most have the right idea• Momentum, denoted Momentum, denoted pp, is mass times velocity, is mass times velocity

p = m·v

• Momentum is a Momentum is a conservedconserved quantity (and a vector) quantity (and a vector)– Often relevant in collisions (watch out for linebackers!)

Winter 2006 7

UCSD Physics 10

Momentum is a vector!!!Momentum is a vector!!! Two Linebackers, as shown, are about to have a Two Linebackers, as shown, are about to have a

collision!!! – Assume they each have 90 kg of masscollision!!! – Assume they each have 90 kg of mass

• What is the total What is the total momentum of both momentum of both linebackers!linebackers!

p = m·v

• Momentum is a Momentum is a conserved quantity conserved quantity (and a vector)(and a vector)– What is the total

momentum of the linebackers after the collisions?

Winter 2006 8

UCSD Physics 10

News headline: Wad of Clay Hits Unsuspecting SledNews headline: Wad of Clay Hits Unsuspecting Sled

• 1 kg clay ball strikes 5 kg sled at 12 m/s and sticks!!1 kg clay ball strikes 5 kg sled at 12 m/s and sticks!!– Determine the initial momentum of the system before the collision.

• Momentum before collision: Momentum before collision: (1 kg)(12 m/s) + (5 kg)(0 m/s) = 12 kg(1 kg)(12 m/s) + (5 kg)(0 m/s) = 12 kg·m/s ·m/s

• Momentum is a conserved quantity (and a vector) Momentum is a conserved quantity (and a vector) – What is the final momentum of the system after the collision?

• Momentum after = 12 kgMomentum after = 12 kg·m/s·m/s– How fast does the total mass of the sled and clay move off together?

• P = mv 12 kgP = mv 12 kg·m/s = (6 kg)· v ; v = (·m/s = (6 kg)· v ; v = (2 m/s2 m/s))

Winter 2006 9

UCSD Physics 10

CollisionsCollisions

• Two types of collisionsTwo types of collisions– Elastic: Energy not dissipated out of kinetic energy

• Bouncy

– Inelastic: Some energy dissipated to other forms• Sticky

• Perfect elasticity unattainable (perpetual motion)Perfect elasticity unattainable (perpetual motion)

Winter 2006 10

UCSD Physics 10

Equal and opposite forces during any collision or Equal and opposite forces during any collision or interaction.interaction.

• Forces are always equal and opposite!!!Forces are always equal and opposite!!!

– Newton’s 3rd Law of Motion!

• Which Canoe experience the greater acceleration?Which Canoe experience the greater acceleration?

• How do the momenta of each canoe compare?How do the momenta of each canoe compare?

– Be sure to consider both MASS and VELOCITY!!!

Winter 2006 11

UCSD Physics 10

Equal and opposite forces, Equal and opposite forces, impulses and changes in impulses and changes in

momentum during a momentum during a collisioncollision

• Impulse = Product of Force and TimeImpulse = Product of Force and Time– J = FΔt

• Impulse is a vector in the same direction as the net force.• Units of impulse are N∙s = kg∙m/s

• Change in momentum = Product of mass and Change in momentum = Product of mass and ΔΔ velocity velocity– Δ p = mΔv

• Change in momentum is a vector. Δ p = m(vf – vi)• Units are N∙s = kg∙m/s

• Impulse = Impulse = Change in momentum : Change in momentum : J = FJ = FΔΔt = t = ΔΔ p = m p = mΔΔvv

Winter 2006 12

UCSD Physics 10

Concept Check?Concept Check?• The magnitude of the initial The magnitude of the initial

momentum is 7 kgm/s and the momentum is 7 kgm/s and the magnitude of the final momentum magnitude of the final momentum is - 5 kgm/s. What is the impulse is - 5 kgm/s. What is the impulse acting on the baseball? acting on the baseball?

• What is the impulse acting on the What is the impulse acting on the bat?bat?

• Impulse on the ball is – 12 kgm/sImpulse on the ball is – 12 kgm/sFFΔΔt = t = ΔΔ p = m p = mΔΔvv

• Impulse on the bat is + 12 kgm/sImpulse on the bat is + 12 kgm/s Impulses are equal and Impulses are equal and opposite!!!opposite!!!

Winter 2006 13

UCSD Physics 10

Problem - ContinuedProblem - Continued

• If the bat is in contact with the If the bat is in contact with the ball for 0.003 sec, what Avg. ball for 0.003 sec, what Avg. force acts on the ball?force acts on the ball?

• J = FJ = FΔΔt, 12kgm/s = F (.003s); F = 4 kNt, 12kgm/s = F (.003s); F = 4 kN

• What is the rate of change in What is the rate of change in the momentum?the momentum?

• F = mF = mΔΔv / t , F = 4 kNv / t , F = 4 kN

• Impulse Impulse GraphsGraphs!!! !!!

Winter 2006 14

UCSD Physics 10

Impulse GraphsImpulse Graphs

• The Area under the curve for a force vs time graph The Area under the curve for a force vs time graph represents the impulse or change in momentum of represents the impulse or change in momentum of the object.the object.

• How would the graph for the baseball appear?How would the graph for the baseball appear?

• Impulse Impulse GraphsGraphs!!! !!!

Winter 2006 15

UCSD Physics 10

Vertical Jump force-time curveVertical Jump force-time curve

• A body weight (BW) line has been added to the graph. The A body weight (BW) line has been added to the graph. The total force acting on the jumper is the ground reaction force total force acting on the jumper is the ground reaction force minus gravitational force (BW). minus gravitational force (BW).

• The shaded regions of the graph sum to produce the impulse The shaded regions of the graph sum to produce the impulse before takeoff and the impulse after landing. The regions before takeoff and the impulse after landing. The regions below the BW line are "negative areas." below the BW line are "negative areas."

Winter 2006 16

UCSD Physics 10

Desk Toy PhysicsDesk Toy Physics

• The same principle applies to the suspended-ball The same principle applies to the suspended-ball desk toy, which eerily “knows” how many balls desk toy, which eerily “knows” how many balls you let go…you let go…

• Only way to simultaneously satisfy energy Only way to simultaneously satisfy energy andand momentum conservationmomentum conservation

• Relies on balls to all have same massRelies on balls to all have same mass

Winter 2006 17

UCSD Physics 10

CollisionsCollisions

• Two types of collisionsTwo types of collisions– Elastic: Energy not dissipated out of kinetic energy

• Bouncy

– Inelastic: Some energy dissipated to other forms• Sticky

• Perfect elasticity unattainable (perpetual motion)Perfect elasticity unattainable (perpetual motion)

Winter 2006 18

UCSD Physics 10

Elastic Collision: Billiard BallsElastic Collision: Billiard Balls• Whack stationary ball with identical ball moving at velocity vcue

8

To conserve both energy and momentum, cue ball stops dead, and 8-ball takes off with vcue

Momentum conservation: mvcue = mvcue, after + mv8-ball Energy conservation: ½mv2

cue = ½mv2cue, after + ½mv2

8-ball

8

The only way v0 = v1 + v2 and v20 = v2

1 + v22 is if either v1 or v2 is 0.

Since cue ball can’t move through 8-ball, cue ball gets stopped.

Winter 2006 19

UCSD Physics 10

Inelastic CollisionInelastic Collision

• Energy not conserved (absorbed into other paths)Energy not conserved (absorbed into other paths)• Non-bouncy: hacky sack, velcro ball, ball of clayNon-bouncy: hacky sack, velcro ball, ball of clay

Momentum before = m1vinitial

Momentum after = (m1 + m2)vfinal = m1vinitial (because conserved)

Energy before = ½m1v2initial

Energy after = ½ (m1 + m2)v2final + heat energy

Winter 2006 20

UCSD Physics 10

QuestionsQuestions

• Twin trouble-makers rig a pair of swings to hang Twin trouble-makers rig a pair of swings to hang from the same hooks, facing each other. They get from the same hooks, facing each other. They get friends to pull them back (the same distance from friends to pull them back (the same distance from the bottom of the swing) and let them go. When the bottom of the swing) and let them go. When they collide in the center, which way do they they collide in the center, which way do they swing (as a heap), if any? What if Fred was pulled swing (as a heap), if any? What if Fred was pulled higher than George before release?higher than George before release?

• A 100 kg ogre clobbers a dainty 50 kg figure A 100 kg ogre clobbers a dainty 50 kg figure skater while trying to learn to ice-skate. If the ogre skater while trying to learn to ice-skate. If the ogre is moving at 6 m/s before the collision, at what is moving at 6 m/s before the collision, at what speed will the tangled pile be sliding afterwards?speed will the tangled pile be sliding afterwards?

Winter 2006 21

UCSD Physics 10

Real-World CollisionsReal-World Collisions

• Is a superball elastic or inelastic?Is a superball elastic or inelastic?– It bounces, so it’s not completely inelastic– It doesn’t return to original height after bounce, so

some energy must be lost

• Superball often bounces 80% original heightSuperball often bounces 80% original height– Golf ball 65%– Tennis ball 55%– Baseball 30%

• Depends also on Depends also on surfacesurface, which can absorb some , which can absorb some of the ball’s energyof the ball’s energy– down comforter/mattress or thick mud would absorb

Winter 2006 22

UCSD Physics 10

Superball PhysicsSuperball Physics

• During bounce, if force on/from floor is purely During bounce, if force on/from floor is purely vertical, expect constant horizontal velocityvertical, expect constant horizontal velocity– constant velocity in absence of forces

– like in picture to upper right

• BUT, superballs often behave contrary to intuitionBUT, superballs often behave contrary to intuition– back-and-forth motion

– boomerang effect

Winter 2006 23

UCSD Physics 10

Angular MomentumAngular Momentum

• Another conserved quantity is Another conserved quantity is angular momentumangular momentum, , relating to relating to rotational inertia:rotational inertia:

• Spinning wheel wants to keep on spinning, Spinning wheel wants to keep on spinning, stationary wheel wants to keep still (unless acted stationary wheel wants to keep still (unless acted upon by an external rotational force, or torque)upon by an external rotational force, or torque)

• Newton’s laws for linear (straight-line) motion Newton’s laws for linear (straight-line) motion have direct analogs in rotational motionhave direct analogs in rotational motion

Winter 2006 24

UCSD Physics 10

Angular MomentumAngular Momentum

• Angular momentum is proportional to rotation Angular momentum is proportional to rotation speed (speed () times rotational inertia (I)) times rotational inertia (I)

• Rotational inertia characterized by Rotational inertia characterized by (mass)(mass)(radius)(radius)22 distribution in object distribution in object

Winter 2006 25

UCSD Physics 10

Angular Momentum ConservationAngular Momentum Conservation

• Speed up rotation by Speed up rotation by tucking intucking in

• Slow down rotation by Slow down rotation by stretching outstretching out

• Seen in diving all the timeSeen in diving all the time

• Figure skaters Figure skaters demonstrate impressivelydemonstrate impressively

• Effect amplified by Effect amplified by moving large masses to moving large masses to vastly different radiivastly different radii

Winter 2006 26

UCSD Physics 10

Do cats violate physical law?Do cats violate physical law?

• Cats can quickly flip Cats can quickly flip themselves to land on themselves to land on their feettheir feet

• If not rotating before, If not rotating before, where do they get their where do they get their angular momentum?angular momentum?

• There are ways to There are ways to accomplish this, by a accomplish this, by a combination of contortion combination of contortion and varying rotational and varying rotational inertiainertia

Winter 2006 27

UCSD Physics 10

For more on falling cats:For more on falling cats:

• Websites: Websites: – www.pbs.org/wnet/nature/cats/html/body_falling.html

• play quicktime movie

– www.exploratorium.edu/skateboarding/trick_midair_activity.html