Momentum

Yet another physics mystery explained

Momentum defined

• Momentum = mass X velocity• Symbol for momentum = “p”• Symbol for mass= “m”• Symbol for velocity= “v”• So p = mv

Let’s think about momentum

• Momentum is made up of 2 quantities: mass and velocity.

• They multiply together to give the momentum.

• Let’s see if we can rank some situations in order of increasing momenta (“momenta” is the plural of “momentum”).

Ranking Momenta

• 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

• If the momentum of an object changes, that means that at least one of these quantities changes– Its mass– Its velocity

• Usually, mass doesn’t change, so it’s most of the time it is the velocity that does

• Recall that Δv/Δt = acceleration

Impulse

• What provides this acceleration?– FORCE

• Let’s say you provide a certain force for a short period of time. You will produce a certain change in momentum.

• What if you provide that same force for longer? You should produce a greater change in momentum.

Impulse

• Example we all know and love: Gravity• If we let something free fall under gravity (no

air resistance) for 3 sec, it’s change in momentum is less than if we let it free fall for 6 seconds.

Impulse defined

• We define “impulse” as: Force X the length of time it is applied.

• Impulse = Ft• Impulse is also equal to the change in

momentum, so: Ft = mv

So what does this mean?

• 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

So what does this mean?

• 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

Which would you rather?

• 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?

Bungee Jumping

Bouncing

• Show of hands: When an object bounces upon landing, is the impulse greater or smaller than when it just sticks?

• Support your answer.

Bouncing: the actual answer

• A greater impulse is needed to make an object bounce than to land and stick.

• Let’s do math:– If a 5 kg object goes from 5 m/s to zero m/s, its

impulse is (5kg)(0 m/s – 5 m/s) = -25 kg m/s– If a 5kg object bounces up at 5 m/s instead of

staying at zero, then impulse is (5kg)(-5m/s – 5m/s) = -50 kg m/s

– In other words, going from V to –V makes the impulse twice as large as going from V to zero

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…?

Collisions (now the REALLY fun part)

• In the absence of external forces, a collision obeys the law of conservation of momentum

• I.E., the total momentum AFTER the collision is the same as the total momentum BEFORE the collision

• There are two kinds of collisions: – Elastic– Inelastic

Collisions Continued (how alliterative…)

• Elastic Collision: a collision in which no energy is lost– In the real world, how is energy lost in a collision?– Sound is generated– Heat is generated– Objects are deformed

• In a perfectly elastic collision, we pretend that these things don’t happen. In the real world this is almost never true, but we can make approximations and pretend.

More collisions• Inelastic collision: a collision in which energy is

NOT conserved (so there is sound/heat/deformation/screaming)

• Even in an inelastic collision, MOMENTUM IS STILL CONSERVED

• Completely inelastic collision: a collision in which the two objects stick together– Examples: – Train cars coupling– Throwing a lump of clay at a student and having it

stick to their head

Sample problem 1 for some lucky student to solve

• Pool ball A (mass 0.1kg) is moving to the right at 2 m/s and hits pool ball B (same mass), which is initially at rest. If pool ball B moves on to the right at 2 m/s, what is the final velocity of pool ball A?

• Set up the problem: PIA + PIB = PFA + PFB

• (0.1kg)(2m/s) + 0 = (0.1kg)(VFA) + (0.1kg)(2m/s)

• 0.2 kg m/s = 0.2 kg m/s + (0.1kg)(VFA)

• So VFA = 0 m/s

Sample problem 2 for some lucky student to solve

• A lump of red clay (1 kg) is moving at 3 m/s to the right and strikes (and sticks to) a lump of blue clay (1kg) moving 3 m/s to the left. What is the final velocity of the system?

• Total PI = Total PF

• (1kg)(3m/s) + (1kg)(-3m/s) = (1kg + 1kg)(VF)• 3 kg m/s – 3 kg m/s = 0 kg m/s• So 0 kg m/s = (2kg)(VF), so VF = 0 m/s

Sample problem 3 for some lucky student to solve

• A railroad car full of Lady Gaga CD’s (mass 10,000 kg) is moving at 5 m/s and then collides and couples with a railroad car full of Lil’ Wayne CD’s (mass 10,000kg), which is initially at rest. What is the final momentum of the system? What is the final velocity of the system?

• Total PI = Total PF , so PF = (10,000kg)(5m/s) = 50,000 kg m/s

• (10,000kg)(5m/s) + 0 = (20,000kg)(PF)

• (50,000 kg m/s) / (20,000kg) = 2.5 m/s = PF

Sample problem 4 for some lucky student to solve

• Car A (mass 1000 kg) slides to the right on frictionless ice at 12 m/s and hits car B (mass 1200 kg), initially at rest. After the collision, car A moves to right at 3 m/s. How fast does car B move?

• (1000kg)(12 m/s) + 0 = (1000kg)(3m/s) + (1200kg)(VFB)

• 12,000 kg m/s = 3000 kg m/s + (1200kg)(VFB)

• 9000 kg m/s = (1200kg)(VFB)• (9000 kg m/s) / (1200kg) = 7.5 m/s