Momentum & Impulse

Impulse

Impulse = the average force applied to an object multiplied by the time over which the force acts

• vector quantity (same direction as the force)• unit is a N s (Newton-second)

• J = Impulse• F = Force• t = time

J = F Δt

Impulse Problem

Steve hits a baseball and gives it an impulse of 100 N s. His bat is in contact with the ball for 1.5 ms.

• How much force did he use?

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J = 100 Ns

t =1.5 ms = 1.5x10-3 s

F = ?

J=FΔt

F=J/ Δt= 100/1.5x10-3

F = 6.67 x 104 Nhttp://www.sapulpascrappers.com/images/scrapper_batter_animated.gif

Momentum

Momentum = an object’s mass

times its velocity• vector quantity (same direction as velocity)• unit is kg m/s (kilogram-meter per second)

• p = momentum • m = mass• v = velocity

 

p=mv

http://www.physicsclassroom.com/Class/momentum/u4l1a.html

“inertia in motion”

Momentum Problems

A baseball of mass 0.10 kg is moving at 35 m/s. Find the momentum of the baseball.

m = 0.10 kg

v = 35 m/s

p = ?

p = mv = (.10)(35)

p = 3.5 kg m/s

Momentum Problems

A compact car, mass 725 kg, is moving at 25 m/s. Find its momentum. At what velocity is the momentum of a larger car, mass 2175kg equal to that of the smaller car?

http://www.3dsplinedesign.com/Portfolio/yugo-.jpg

m=725 kg

v = 25 m/s

p = ?

p = mv

p = 725(25)

p = 18,125 kg m/s

p2= p1

p2 = 18,125 kg m/s

m2 = 2175 kg

p = mv

v = p/m = 18,125

2175v = 8.3 m/s

The Relation Between Force and Momentum

• Newton’s 2nd Law

• Recall the definition of acceleration

• Substituting,

• Remember momentum

• Therefore,

• So,

• F Δt = Impulse (J)• J = Δp

F=ma

t

va

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vmF

Δp=m Δv

t

pF

Δp=F Δt

Note: This derivation is the basis for the impulse-momentum theorem

Impulse-Momentum Theorem

Impulse-Momentum Theorem = the impulse of an object equals the change in momentum of that object

FΔt = mΔv = m vf – m vo

J=Δp

http://www.physicsclassroom.com/Class/momentum/u4l1a.htm

Impulse-Momentum Theorem • the impulse of an object equals the change

in momentum of that object.

http://www.physicsclassroom.com/Class/momentum/u4l1a.htm

FΔt = mΔv = m vf – m vo

J=Δp

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

Examples of Impulse Increasing

Momentum:• apply greatest force

possible (large F)• extend time of

contact (t)• ex. Baseball player

and golfer as they follow through. (they increase t)

Changing Force1. over a long time:• want to lessen force • extend impact time,

reduce impact force• ex. Airbags in cars

(increases t)2. over a short time:• the impact force is

large• ex. Breaking blocks

by using karatehttp://www.hk-phy.org/contextual/mechanics/mom/impulse/4-01.gifhttp://www.shihanroman.com/Breaking_ShiRoman.htm

Movie Punches• How does a stuntman take a punch?

Why?

http://www.physicsclassroom.com/Class/momentum/U4L1c.html

Clicker Understanding

Raindrops keep falling …• What would do more damage to your car a

raindrop or hailstone of equal masses hitting it? Why?

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

A. RaindropB. HailstoneC. Same

Impulse Momentum Problem

. A snowmobile has a mass of 2500 kg. A constant force is exerted on it for 60 sec. The snowmobile’s initial velocity is 6m/s and its final velocity 28m/s. What is the change in momentum? What is the force exerted on it?

m =2500 kg

t = 60 s

vi= 6 m/s

vf = 28 m/s

Δp = m Δv= m(vf-vo)

Δp = 2500(28-6)

Δp = 55,000 kg m/s

Momentum Changing Forces• If the force acting on the

body is not constant we can write

• Δp = average force × Δt

• Suppose the force acting on a body varies as shown below.

• During the first three seconds the change in momentum was

• During the next four seconds the change in momentum was

• Δp2 = 8 × 6 = 48 Ns• So the total change in momentum was 60 Ns

tFF

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p of 1232

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Notice that the change in momentum can be found by calculating the area under the graph of force against time.

Graphical Interpretaion of Impulse• J = Impulse = area under

the force curve

Clicker Understanding• Two 1-kg stationary cue balls are struck by cue

sticks. The cues exert the forces shown. Which ball has the greater final speed?

A. Ball 1B. Ball 2C. Both balls have the same final speed

Conservation of Momentum • Momentum is conserved• The total momentum before collision or

interaction equals the total momentum after a collision.

• The change in momentum of one object is equal and opposite to the change in momentum of the other object in a collision.

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(Newton’s 3rd Law)

po=pf

Δp1=-Δp2

Conservation of Momentum

Assumption:

No net external forces are acting on an isolated system of objects, the total vector momentum of the system remains constant.

Derive from Newton’s 2nd & 3rd Laws

Derivation:

Conservation of Momentum• Initial Conditions

– ma with va and mb with vb

• After the collision the velocities change to ua and ub

• Assume F is the force that object A experiences for a time Δt during the collisions

• Newton’s 2nd Law

• Newton’s 3rd Law• Force experienced by B is -F

• thus,

pinitial = pfinal

Conservation of Momentum

Derivation:

Conservation of Momentum

Conservation of Momentum

http://www.physicsclassroom.com/Class/momentum/U4L2c.html

Clicker Understanding

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

A. It takes less time to stop the ping-pong ball.

B. Both take the same time

C. It takes more time to stop the ping-pong ball.

Clicker Understanding

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the distances needed to stop them compare?

A. It takes a shorter distance to stop the ping-pong ball.

B. Both take the same distance

C. It takes a longer distance to stop the ping-pong ball.

CollisionsConservation of Momentum always occursTypes of collisions:Elastic:• a collision in which kinetic energy is conserved. Inelastic:• a collision in which kinetic energy is NOT

conserved.• some K.E. is converted to other forms of energy. Totally Inelastic (Perfectly Inelastic)

after collision the two objects stick together and move with the same velocity.

Types of collisions

Objects collide and bounce off each other (Elastic and Inelastic Collisions)

Solving,

Conservation of Momentum Examples

Glider A of mass 0.355 kg moves along a frictionless air track with a velocity of 0.095 m/s. It collides with glider B of mass 0.710 kg moving in the same direction at a speed of 0.045 m/s. After the collision, glider A continues in the same direction with a velocity of 0.035 m/s. What is the velocity of glider B after the collision?

Totally Inelastic Collision

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

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Types of collisions

Totally Inelastic collision = 2 objects collide and stick together, moving off together

• Usually one of the objects is initially at rest

• Solving,

Swallowing: a Fish Story

What is the final velocities of the two fish?

http://www.physicsclassroom.com/Class/momentum/U4L2dd.html

Totally Inelastic Example• Calculate the final velocity of the engine

and flatbed car after the collision.

http://www.physicsclassroom.com/mmedia/momentum/dft.html

Conservation of Momentum Examples

A 0.105 kg hockey puck moving at 48 m/s is caught by a 75kg goalie at rest. With what speed does the goalie slide on the ice?

Inelastic vs Totally Inelastic CollisionsTotally Inelastic:• Momentum Conserved• KE Not Conserved

• Objects stick together after collision.

• Objects have the same velocity after the collision.

Inelastic:• Momentum Conserved• KE Not Conserved

• Objects move separately after collision.

• Objects have different velocities after the collision.

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

Similarities

Differences

Elastic CollisionsIf initially one object is moving and the other is stationary

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Types of Collisions

Copywrited by Holt, Rinehart, & Winston

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Explosions

Is momentum conserved during explosions?

http://www.glenbrook.k12.il.us/gbssci/phys/Class/momentum/u4l2e1.gif

Types of collisions

Explosions- both objects start at rest and move in opposite directions

Solving,

Clicker UnderstandingA person attempts to knock down a large wooden bowling pin by throwing a ball at it. The person has two balls of equal size and mass, one made of rubber and the other of putty. The rubber ball bounces back, while the ball of putty sticks to the pin. Which ball is most likely to topple the bowling pin?

A. The rubber ball

B. The ball of putty

C. Makes no difference

D. Need more information

Identify the type of Collision

http://www.physicsclassroom.com/mmedia/momentum/cthoe.html

Identify the type of Collision

http://www.physicsclassroom.com/mmedia/momentum/cthoe.html

Identify the type of Collision

http://www.physicsclassroom.com/mmedia/momentum/cthoe.html

Clicker UnderstandingThink fast! You’ve just driven around a curve in a narrow, one-way street at 25 mph when you notice a car identical to yours coming straight toward you at 25 mph. You have only two options: hitting the other car head-on or swerving into a massive concrete wall, also head on. In the split second before the impact, you decide to

A. Hit the other car.

B. Hit the wall.

C. Hit either one – it makes no difference.

D. Consult your notes. Myth Busters

ReviewIn a head-on collision:

Which truck will experience the greatest force?Which truck will experience the greatest impulse?Which truck will experience the greatest change in momentum?Which truck will experience the greatest change in velocity?Which truck will experience the greatest acceleration?Which truck would you rather be in during the collision?

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

2-D Collisions• Since momentum is a vector, Conservation of

momentum can be extended to two dimensional scenarios.

• The x and y components of momentum as well as its total magnitude must be the same before and after the collision.

Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

2-D Collisions

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Example Problem:Two pucks on an air hockey table collide. Puck A

has a mass of 0.025 kg and is moving along the x axis with a speed of 5.5 m/s. It collides with puck B, which has a mass of 0.050 kg and is initially at rest. After the collision the two puck fly apart as shown. Find the final speed of puck A and B.

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