Impulse and Momentum and Collisions and Stuff

We will now quantify exactly how it hits the fan.

Momentum

• Momentum = product of mass X velocity• Symbol for momentum is “p” (don’t ask me why)– So, p = mv

• Momentum is also a vector and points in the direction of velocity

• Units for mass = kg, units for velocity = m/s• So, units for momentum = kg•m/s– Sadly, there’s no other, more convenient unit for it

• Plural is ‘momenta’

A closer look at momentum

• Think of momentum as being how easy/hard it is to get something to stop.

• Remember, p = mv• So if mass or velocity is small, it’s likely that

the object has a small momentum, unless the other of the two is very large

• What would be the momentum of an object at rest?

5 Scenarios• Let’s look at the momenta of….• A car at rest• Mass = 1000 kg, velocity = 0 m/s• P = mv = (1000kg)(0m/s) = 0 kg m/s

• A bumblebee in flight (low mass, low velocity)• Mass = 0.05 kg, velocity = 2 m/s• P = mv = (0.05kg)(2m/s) = 0.1 kg m/s

• Projectile from BB gun (low mass, high velocity)• Mass = 0.01 kg, velocity = 150 m/s • P = (0.01 kg)(150 m/s) = 1.5 kg m/s

5 Scenarios continued

• A giant land tortoise (large mass, low velocity)• Mass = 300 kg (I looked it up!), velocity = 0.5 m/s• P = (300kg)(0.5m/s) = 150 kg m/s

• Tony Stewart at the Brickyard (large mass, high velocity)– Mass = 1000 kg, velocity = 90 m/s– P = (1000kg)(90 m/s) = 90,000 kg m/s

Understanding momentum

• Which is easier to stop…• A slow moving baseball or a fast moving

baseball?• A chihuahua racing down the hall towards you

or Mr. Barbini racing down the hall towards you?

• A speeding Volkswagen beetle or a speeding dump truck?

I need a volunteer

• Rank these momenta from lowest to highest– A fast flying bee– The earth in orbit around the sun– A parked garbage truck– A slowly flying bee– Your grandmother driving down the street in her

1959 Edsel– An oil tanker sailing the seven seas. – Tony Hawk grinding on a rail

Impulse

• Impulse = Product of Force and how long that force acts. – Impulse = F∆t

• Impulse is a vector. It acts in the same direction as the force

• Units of force = Newtons, units of time = seconds. So, units for impulse = Newton•Seconds

Okay, but what does that really mean?

• Let’s say you have a force of 10 N. You apply this force to a hockey puck on (frictionless) ice for 2 seconds.

• Impulse = (10N)(2 sec) = 20 Newton seconds• Now, let’s say you apply this same force for 4

seconds• So the impulse now = 40 Newton seconds

Impulse-momentum theorem

• Remember the work-energy theorem? This is kind of like that.

• Impulse = change in momentum• In general, the mass of the object(s) in

questions stays constant• ∆p = pF – pI = mvF – mvI

• So F∆t = mvF – mvI

Well, Duh…

• Let’s think about 2 situations:• Situation 1: pushing someone in a rolling chair– If you push on them with a certain force for two

different times, the longer time will result in a greater change in momentum

– So if the chair was at rest to begin with, it will end up going faster at the end for the longer time the force was applied

Less Duh

• Situation 2: crashing your 1,000 kg car• Let’s say you are driving along at 45 m/s (about

96 mph) and you crash into one of two objects: a solid wall or a series of water-filled plastic bins (like they have on the highway)

• If your car goes from 45 m/s to 0 m/s, you have a change in momentum of:

• P = mv, so momentum = (1000kg)(45m/s) = 45,000 kg•m/s

The car crash continued

• Recall that impulse = change in momentum, so Ft = mv = 45,000 kg•m/s

• So, Force X time = 45,000 kg•m/s• Now, would you rather have that force spread

out over a long time, or over a short time?

Compare stopping times

• Change in momentum is 45,000 kg•m/s• For a stopping time of 5 sec (slamming on your

brakes): (F)(5 sec) = 45,000 kg•m/s– So F = 9,000 N

• For a stopping time of 1.5 sec (smashing into the plastic water bins): (F)(1.5 sec) = 45,000 kg•m/s– So F = 30,000 N

• For a stopping time of 0.2 sec (smashing into the concrete wall): (F)(0.2 sec) = 45,000 kg•m/s– So F = 225,000 N

Let’s look at those numbers more closely…

• Remember we have a 1,000 kg car and F = ma• If F = 9000N, then accel = 9 m/s (just under

acceleration due to gravity)• If F = 30,000N, then accel = 30 m/s (just over

3X acceleration due to gravity)• If F = 225,000N, then accel = 225 m/s (about

23X the acceleration due to gravity)• So, which is the most survivable?

Let’s look at some examples

• Hitting a baseball (mass = 0.14 kg)• Let’s say that a baseball is travelling to the left (negative

velocity) at -38m/s. After being hit, it moves to the right at 58 m/s. What is the impulse applied to the ball? If the contact time was 1.6X10-3 sec, what was the force?

• Impulse = ∆p = pF – pI = mvF – mvI

• Impulse = (0.14kg)(58m/s) – (0.14kg)(-38m/s)• Impulse = 13.4 kg m/s• Impulse = F∆t, so F = impulse/∆t • Force = (13.4 kg m/s)/(1.6X10-3 sec) = 8400 N

Conservation of Momentum

• In general, momentum is conserved• This means that the momentum at the

beginning is the same as the momentum at the end

• I.e. momentum is not created or destroyed• Remember, symbol for momentum is ‘p’• So PI = ‘initial momentum’ and PF = ‘final

momentum’• Let’s look at some examples

Conservation of Momentum: Example 1

• Cannon and cannonball• What is momentum before

cannon is fired (PI)?• Zero• So what does final

momentum (PF) have to be if momentum is conserved?

• Zerohttp://www.sparknotes.com/testprep/books/sat2/physics/chapter9section3.rhtml

Conservation of Momentum: Example 1• Let’s say that the cannon

has a mass of 1000 kg and Cannonball has a mass of 10 kg

• So total PI = PI of ball + PI of cannon

• PI of ball = (10kg)(0m/s) = 0 kg m/s

• PI of cannon = (1000kg)(0m/s) = 0 kg m/s

Conservation of Momentum: Example 1• Now let’s say that the

cannonball moves to the right at 75 m/s. How fast does the cannon move to the left to conserve momentum?

• PI = PF

• So PF must equal zero

Conservation of Momentum: Example 1• PF must equal zero

• Total PF = PF of ball + PF of cannon

• Remember, moving to the right = positive velocity

• Moving to the left = negative velocity

Conservation of Momentum: Example 1• PF = 0, so • (10kg)(75 m/s) + (1000kg)

(Vcannon) = 0• So -Vcannon =

(10kg)(75m/s) / (1000kg)• Vcannon = -0.75 m/s• If ball moves to the right,

cannon moves to the left, so Vcannon should be negative, which it is

Conservation of Momentum: Example 2

• Newton’s cradle (or “Executive ball clicker” or, more crassly, “Newton’s Balls”)

• How does it work?• If one ball is used, how

many come up on the other side?

• If two are used…?

Now, Dr. Mason, remind us again when conservation of momentum holds…

• Momentum is conserved when there are NO EXTERNAL FORCES ACTING ON THE SYSTEM.

• If any net external force acts, momentum will NOT be conserved.

• Let’s look at a contrived example involving pool balls.

Billiards• Imagine the scenario where one pool ball is stationary

on a pool table. A second pool ball smacks into it. – The ‘system’ is the two pool balls.– Are there any net external forces?

• Now, imagine that just before they hit, a hole opens up underneath them.– The ‘system’ is the two pool balls.– Now are there any net external forces?

• Okay, this time, no sudden trapdoor. Same two pool balls collide.– The ‘system’ this time is just one pool ball.– Now are there any net external forces?

Freight trains example• A freight train is being assembled out of two boxcars. • Car 1 has a mass of 65,000 kg and is moving to the

right at V01 = 0.8m/s. • Car 2 has a mass of 92,000 kg and is also moving to

the right at V02 = 1.3 m/s. • Car 2 collides with car 1 and couples with it. What is

the speed of the train after the coupling?• Momentum is conserved.• (m1 + m2)Vf = m1V01 + m2V02

• Vf = 1.1 m/s

Ballistic Pendulum• A ballistic pendulum is a device that is used to

determine the muzzle velocity of a gun.

http://session.masteringphysics.com/problemAsset/1010989/26/1010989A.jpg

Collisions in 2-D

• Conservation of momentum holds in 2 dimensions as well

• As one may expect, you can treat the X- and Y-components independently – So, Pf = Pi

– And PfX = PiX

– And PfY = PiY