Momentum

Notes (HRW p196)

Momentum

Definition • The measure of how difficult it is to stop a moving

object

• Mass in Action!

Formula • p = mass * velocity, or

• p = mv

Units • kg.m/s

Momentum is a vector!

Mass in Action!

Example

Q: What is the momentum of a 1000 kg Civic traveling at 30 m/s? • p = mass x velocity = m x v

• p = 1000 x 30 = 30,000 kg.m/s

Q: What is the momentum of a 100,000 kg locomotive traveling at 30 m/s? • P = mass x velocity

• P = 100,000 x 30 = 3,000,000 kg.m/s

Q: What is the momentum of a 40,000 ton (40,000,000 kg) oil tanker traveling at 5 m/s? • P = m * v = 40,000,000 * 5

• P = 200,000,000 kg.m/s

Mass in Action!

Impulse

Definition • The product of the force (F) acting on an object and

the duration of the force (t)

Formula • Impulse = F * t

Units • Newtons.seconds (N.s)

Examples • Striking a golf/foot/base ball

• Seatbelts (how do they work?) • Auto safety developments since 50’s

Mass in Action!

Impulse Example

Example: Wall exerts a force of 10,000 N on

the van. The contact time is 0.01 s. What is the

impulse?

Solve: Impulse = F * t = 10,000 * 0.01

• Impulse = 100 N-s

Mass in Action!

Ex. Impulse = Momentum Change

Impulse = change in momentum (final – initial)

Impulse = 0 – mv

Ft = -mv (momentum is a vector!)

F = mv/t (force felt is the momentum/duration of the

applied force)

Mass in Action!

Impulse and Momentum Change

According to Newton’s 2nd Law • The application of a net force on an object will

result in the object accelerating (aka changing velocity)

• F = ma = m(Δv/t) = m(vf – vo)/t, but

• Ft= mvf – mvo = pf – po thus

• Ft (or impulse) = change in momentum

• Ft = Δmv = mΔv

• F = mΔv/t

• F = m(vf – v0)/t (Newton’s 2nd law in momentum terms)

Mass in Action!

Example – Madness!

Mass in Action!

Impulse Example

A 1000 kg Civic is traveling at 30 m/s and accelerates to 40 m/s in 10 seconds. • What is the momentum of the car before accelerating?

• po = m*v = 1000 * 30 = 30,000 kg.m/s

• What is the momentum of the car after accelerating for 10 seconds?

• pf = m*v = 1000*40 = 40,000 kg.m/s

• What is the change in momentum?

• Δp = pf – po = 40,000 – 30,000 = 10,000 kg.m/s

• What is the impulse?

• J = Change in momentum = 10,000 kg.m/s

• What is the net force that causes the change?

• F = change in momentum/time = 10,000/10 = 1,000 N

F*t = change in momentum = mΔv Mass in Action!

Impulse Example

A 1000 kg Civic is traveling at 30 m/s

and hits a lamp post.

• What is the momentum of the car while

moving?

• po = m*v = 1000*30 = 30,000 kg.m/s

• What is the momentum of the car after hitting

the post?

• pf = m*v = 1000*0 = 0 kg.m/s

• What is the change in momentum?

• Δp = pf – po = 0 – 30,000 = -30,000 kg.m/s

Mass in Action!

Impulses and Contact Time

How does momentum of the vehicle relate to the impulse in the 2

scenarios below?

Force is spread over a longer duration

Force is spread over a shorter duration!

Mass in Action!

Summary: Momentum - Impulse

Interrelated

• Momentum

• Impulse

• Change in momentum

IMPULSE

Force * time

F*t

Change in

momentum

F*t = m(vf – vo)

Momentum

Δp = mvf - mvo

Mass in Action!

Summary

Mass in Action!

Practice

A stationary 0.12 kg hockey puck is hit

with a force that lasts for 0.01 seconds

and makes the puck move at 20 m/s.

With what force was the puck hit?

Impulse = Change in momentum

• Ft = mΔv = m(vf – vo)

• F = mΔv/t

• F = 0.12 (20 – 0)/0.01 = 240 N

Mass in Action!

Practice

If a 5kg object experiences a 10 N force

for 0.1 second, what is the change in

momentum of the object?

Solve

• Change in momentum = impulse

• Impulse = F*t

• Change in momentum = F*t = 10*0.1 = 1N-s

Mass in Action!

Practice

A 50 kg driver of a sports car is traveling at

35 m/s when she hits a large deer. She

strikes the air bag/seatbelt combination that

brings her body to a stop in 0.5 seconds.

• What average force does the bag/belt exert on

her?

Solve: m=50, vo=35, vf=0, t=0.5, F=?

• F = m*(vf-vo)/t

• F = 50*(-35)/0.5 = -3500 N

Mass in Action!

Practice (continued)

What if the driver in the previous

example was not wearing a seatbelt and

there were no airbags, and the

windshield stopped her head in 0.002 s.

• What is the average force on her head?

Solve

• F = 50*-35/0.002 = -875,000 N!!!

Mass in Action!

Conservation of Momentum

Momentum Before firing = p(rifle) + p(bullet) = 0

Momentum After firing = p(rifle) + p(bullet) = 0

After firing, the opposite momenta cancel – direction is

important in vector arithmetic!

Sum of the momenta of all elements before the event = Sum of

the momenta of all elements after the event

Mass in Action!

Conservation of Momentum –

Collisions (aka events)

2 basic types of collisions for analysis • Elastic

• Bodies collide and bounce apart – no energy loss

• Bowling ball/pin

• Pool

• Inelastic • Bodies collide and stick together – some energy

transformation into heat

• Auto rear-ender

• Picking up an object (combining)

Real world – a bit of both!

Mass in Action!

Momentum Table

Before vs After

BEFORE Object 1 Object 2 Total Momentum

Mass (m) m1 m2

Velocity (v) v1 v2

Momentum m1v1 m2v2 m1v1 + m2v2

AFTER Object 1 Object 2 Total Momentum

Mass (m) m1 m2

Velocity (v) v1 v2

Momentum m1v1 m2v2 m1v1 + m2v2

Equate the total momentum (before) with the total momentum (after) to

solve!

Mass in Action!

Practice using table

A 1000 kg Honda @ 30 m/s collides (head-on

glancing blow) with a 2000 kg Camry @ 20

m/s. If the Honda continues @ 20 m/s,

• what is the speed of the Camry after the impact?

Solve: Honda, Camry, collision

• Total momentum (before) = Total momentum (after)

• phonda + pcamry = phonda + pcamry

• 1000*30 + 2000*(-20) = 1000*20 + 2000*v

• 30,000 – 40,000 = 20,000 + 2000v

• v = -15 m/s (negative direction)

Mass in Action!

Practice using table

A 200 kg astronaut leaves the Shuttle for a space walk and while stationary his tether breaks. To return to the craft he throws a 2 kg hammer @ 3m/s directly away from the shuttle. • What is his returning speed?

Solve: astronaut, hammer, throw • pastro + phammer (before) = pastro + phammer (after)

• 200*0 + 2*0 = 200*v + 2*3

• v = -200/6 = -0.03 m/s

Mass in Action!

Practice using table

A 500 kg bumper car @ 5 m/s runs into the

back of a similar car @ 2 m/s. If the 2nd car

bounces forward at 4 m/s,

• What is the speed of the 1st car?

Solve: bumper car #1, #2, collision

• (p1 + p2)before = (p1 + p2)after

• 500*5 + 500*2 = 500*v + 500*4

• 2500 + 1000 = 500v + 2000

• v = 3 m/s

Mass in Action!