Momentum Momentum

A.K.A. The difference between A.K.A. The difference between moving and standing still.moving and standing still.

DefinitionDefinitionMathematical

Momentum Momentum = Mass (kg) x Velocity (m/s)= Mass (kg) x Velocity (m/s)OrOr

p=mvp=mv

The units for momentum are kgm/sThe units for momentum are kgm/s

Verbal

Momentum is “inertia in motion”. Momentum is “inertia in motion”.

Remember Newton’s 1Remember Newton’s 1stst law. It’s law. It’s analogous to Inertia.analogous to Inertia.

Momentum = mass x Momentum = mass x velocityvelocity

• MomentumMomentum is a true measure of is a true measure of how difficult it is to stop how difficult it is to stop something.something.

– A charging hippo can do some damage, a A charging hippo can do some damage, a hippo charging hippo charging twicetwice as fast can do as fast can do twicetwice the damage.the damage.

– Calculating momentum is easy, just find the Calculating momentum is easy, just find the mass and the velocity and multiply.mass and the velocity and multiply.

p = mvp = mv– Notice that mass and velocity both affect Notice that mass and velocity both affect

momentum momentum equallyequally..

ImpulseImpulse – where momentum – where momentum comes from!comes from!

• Only a force can “give” something momentum (or take it away). – Lets say you are a member of a bobsled team. You push

the sled to speed it up. The longer the you push the sled the greater the velocity and the greater the momentum you give it so time is also a factor.

– Or think about the airbags in your car. They give you more time to slow down so less force is applied to your body.

• So… a force applied for a certain time leads to a change in momentum.

• Δp = (Force) x (time) This is called an impulse (J)J = Favg ∆ t

• Impulse is a change in momentum (∆mV)∆(mV) = FtJ = m∆V = Favg ∆ t

• The units for impulse are the same as momentum (kgm/s)

Constant Force IMPULSEConstant Force IMPULSE

t

F

J = Favg tJ = Favg t

Impulse:Impulse J is a force F acting for a small time interval t.

Example 1:Example 1: The face of a golf club The face of a golf club exerts an average force of exerts an average force of 4000 N4000 N for for 0.002 s0.002 s. What is the impulse . What is the impulse imparted to the ball?imparted to the ball?

t

F

J = F tJ = F tImpulse:

J = (4000 N)(0.002 s)

J = 8.00 kg m/sJ = 8.00 kg m/s

The unit for impulse is the kg m/s

Impulse from a Varying Impulse from a Varying ForceForce

Normally, a force acting for a short interval is not constant. When force is not constant you need to look at a Force vs Time graph.

Impulse is the area of a Force vs Time graph.

This is similar to the area of a Force vs

Position Graph

Example 2:Example 2: Two flexible balls Two flexible balls collide. The ball collide. The ball BB exerts an average exerts an average force of force of 1200 N1200 N on ball on ball AA. How long . How long were the balls in contact if the were the balls in contact if the impulse is impulse is 5 N s5 N s??

t = 0.00420 st = 0.00420 s

The impulse is The impulse is negativenegative; the force on ; the force on ball A is to the left. Unless told ball A is to the left. Unless told otherwise, treat forces as otherwise, treat forces as average average forcesforces..

BAavgJ F t

-5 N s

-1200 Navg

Jt

F

Impulse Changes VelocityImpulse Changes VelocityConsider a mallet hitting a

ball:F

; f ov vF ma a

t

; f ov v

F ma at

f oF t mv mv f oF t mv mv 0fv vF m

t

Impulse = Change in “mv”Impulse = Change in “mv”

Momentum DefinedMomentum DefinedMomentum p is defined as the product of mass and velocity, mv. Units: kg m/s

p = mvp = mv Momentum

m = 1000 kg

v = 16 m/s

p = (1000 kg)(16 m/s)

p = 16,000 kg m/sp = 16,000 kg m/s

Impulse and MomentumImpulse and Momentum

Impulse = Change in momentumImpulse = Change in momentum

F t = mvf - mvoF t = mvf - mvo

t

F mvA force F acting on a ball for a time t increases its momentum mv.

Example 3:Example 3: A A 50-g50-g golf ball leaves golf ball leaves the face of the club at the face of the club at 20 m/s20 m/s. If . If the club is in contact for the club is in contact for 0.002 s0.002 s, , what average force acted on the what average force acted on the ball?ball?

t

F mv

Given:Given: m = 0.05 kg; vo = 0;

t = 0.002 s; vf = 20 m/s+

Choose right as Choose right as positive.positive.

F t = mvf - mvo

F (0.002 s) = (0.05 kg)(20 m/s)

Average Force: F = 500 NF = 500 N

0

Vector Nature of Vector Nature of MomentumMomentum

Consider the change in momentum of a ball that is dropped onto a rigid

plate:

vo

vf

A 2-kg ball strikes the plate with a speed of 20 m/s and rebounds with a speed of 15 m/s. What is the change in momentum?

+

p = mvf - mvo = (2 kg)(15 m/s) - (2 kg)(-20 m/s)p = 30 kg m/s + 40 kg m/s

p = 70 kg m/s

p = 70 kg m/s

Directions Are EssentialDirections Are Essential

1. Choose and label a positive direction.1. Choose and label a positive direction.

+ vf

v0

vvff – v – v00 = = (10 m/s) – (-30 (10 m/s) – (-30

m/s)m/s)

40 m/sv

2. A velocity is positive 2. A velocity is positive when with this direction when with this direction

and negative when and negative when against it.against it.Assume Assume vv00 is 30 m/s is 30 m/s

to the left and to the left and vvff is is 10 m/s to the right.10 m/s to the right. What is the change What is the change in velocity in velocity v?v?

vvff = +10 m/s= +10 m/s

vv00= -30 m/s= -30 m/s

Example 4:Example 4: A A 500-g500-g baseball moves baseball moves to the left at to the left at 20 m/s20 m/s striking a bat. striking a bat. The bat is in contact with the ball for The bat is in contact with the ball for 0.002 s0.002 s, and it leaves in the opposite , and it leaves in the opposite direction at direction at 40 m/s40 m/s. What was . What was average force on ball?average force on ball?

40 m/s

t

F20 m/s

m = 0.5 kg

+

- +

F t = mvf - mvo

F(0.002 s) = (0.5 kg)(40 m/s) - (0.5 kg)(-20 m/s)

vo = -20 m/s; vf = 40 m/s

Continued . . .

Example Continued:Example Continued:

40 m/s

t

F20 m/s

m = 0.5 kg

+

-+

F t = mvf - mvo

F(0.002 s) = (0.5 kg)(40 m/s) - (0.5 kg)(-20 m/s)

F(0.002 s) = (20 kg m/s) + (10 kg m/s)

F(0.002 s) = 30 kg m/s F = 15,000 NF = 15,000 N

Summary of Formulas:Summary of Formulas:

Momentum p = mvMomentum p = mv

Impulse J = FavgtImpulse J = Favgt

Impulse = Change in momentumImpulse = Change in momentumImpulse = Change in momentumImpulse = Change in momentum

F t = mvf - mvoF t = mvf - mvo

Impulse and MomentumImpulse and Momentum