Momentum

Unit 6: Momentum & Impulse

Which is more difficult to stop:

• A tractor-trailer truck barreling down the highway at 35 meters per second OR
• A small two-seater sports car traveling the same speed?

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Momentum

• Inertia in motion 🡪 a measure of how hard it is to stop a moving object
• Depends on mass and velocity
• How much stuff is moving
• How fast the stuff is moving

p = mv

m = mass (kg)

p = momentum

(kg∙m/s)

v = velocity (m/s)

Momentum

• Is a vector 🡪 has direction
• Momentum is directly proportional to mass and velocity
• double mass = double momentum​
• double velocity = double momentum

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Sample Problems

1. A 0.02 kg pencil is moving at 25 m/s. What is its momentum?

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• The pencil would have the same momentum as a 0.14 kg baseball traveling at...

m = 0.02 kg

v = 25 m/s

p = ?

p = mv

p = (0.02)(25)

p = 0.5 kg∙m/s

m = 0.14 kg

p = 0.5 kg∙m/s

v = ?

p = mv

0.5 = 0.14v

v = 3.57 m/s

Change in Momentum

• Δp = m(v_f – v_i) 🡪 or Δp = mΔv
• Compare the change in momentum to the beanbag bear and the rubber ball

Bear

Δp = m(v_f – v_i)

Δp = 1(0 – -4)

Δp = (1)(4)

Δp = 4 kg∙m/s

Ball

Δp = m(v_f – mv_i)

Δp = 1(4 – -4)

Δp = (1)(8)

Δp = 8 kg∙m/s

Total Momentum 🡪 p_total = p₁ + p₂+ …

At a city park a person throws some bread into a duck pond. Two 4.0 kg ducks and a 7.6 kg goose swim towards the bread from opposite directions. If the ducks swim at 1.1 m/s and the goose swims at 1.3 m/s, find the magnitude and direction of the total momentum of the three birds.

p_total = p₁ + p₂ + p₃

p_total = mv₁ + mv₂ + mv₃

p_total = (7.6)(1.3) + (4)(-1.1) + (4)(-1.1)

p_total = 9.88 + (-4.4) + (-4.4)

p_total = 9.88 – 8.8

p_total = 1.08 kg∙m/s

6.1 Lesson Check

Impulse

Unit 6: Momentum & Impulse

Impulse = Change in Momentum

But wait – we already learned that Δp = m(v_f – v_i)

An impulse is what causes that change in momentum to occur

Impulse: change in momentum

J = F△t

F = force (N)

J = impulse

(N∙s)

t = time (s)

Sample Problem 1

You push on a 22 kg shopping cart with a force of 6.5N. If you push for 1.9 s, what is the magnitude of the impulse you deliver to the cart?

Givens

m = 22 kg

F = 6.5 N

t = 1.9 s

J = FΔt

J = (6.5)(1.9)

J = 12.35 N∙s

Impulse and Momentum

• An impulse changes an object’s momentum
• To stop a moving object, it is necessary to apply a force against its motion for a given period of time
• As a force acts upon the object for a given amount of time, the object's velocity is changed, which in turn changes its momentum

A force acting for a given amount of time will change an object's momentum 🡪 an unbalanced force always accelerates an object – either speeding it up or slowing it down

• If the force acts opposite the object's motion, it slows the object down
• If a force acts in the same direction as the object's motion, then the force speeds the object up
• Either way, a force will change the velocity of an object, which changes its momentum

Impulse-Momentum Theorem

The impulse-momentum theorem states:

if J = F△t

and impulse = change in momentum

then F△t = m△v

and F△t = m(v_f – v_i)

OR F△t = mv_f – mv_i

Sample Problem 2

Three friends apply a force of 305 N to push a stalled (not moving) car. The mass of the car with the driver inside is 1360 kg. How fast is the car moving after 12 s of pushing?

Givens

F = 305 N

V_i = 0 m/s

m = 1360 kg

t = 12 s

FΔt = mΔv

FΔt = m(v_f – v_i)

(305)(12) = (1360)(v_f - 0)

3660 = 1360v_f

v_f = 2.69 m/s

Increasing the time of impact decreases the force

Compare the outcome of dropping an uncooked egg onto a plate vs a foam pillow

• Both eggs experience the same impulse because they had the same initial and final momentum
• The egg that landed on a plate experienced a larger force and shorter time interval to come to rest compared to the egg that landed on the foam pillow
• The foam pillow extends the time interval over which the egg came to rest which decreases the force exerted

Increasing the time of impact decreases the force

Padding in helmets, air bags, and race car tracks all reduce the force exerted by extending the time interval over which an object comes to rest

This Photo by Unknown Author is licensed under CC BY

The Amazing Spider-Man?

The Amazing Spider-Man?

Spider-Man saves falling people using his webs. Gwen Stacey, however, is not so lucky, as her neck breaks when Spiderman catches her. Explain what would have to happen for Gwen Stacey to be saved.

If Spiderman had shot out his web to catch Gwen sooner, then she would have taken more time to get to the bottom, so there would have been less force and her neck would not have broken.

6.2 Lesson Check