So What’s Momentum ?

• Momentum = mass x velocity
• Momentum is a measure of inertia in motion
• This can be abbreviated to : . momentum = mv

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• Or, if direction is not an important factor : . . momentum = mass x speed

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• So, A really slow moving truck and an extremely fast roller skate can have the same momentum.

Question :

• Under what circumstances would the roller skate and the truck have the same momentum ?

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• The roller skate and truck can have the same momentum if the ratio of the speed of the skate to the speed of the truck is the same as the ratio of the mass of the truck to the mass of the skate.

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• A 1000 kg truck moving at 0.01 m/sec has the same momentum as a 1 kg skate moving at 10 m/sec. Both have a momentum of 10 kg m/sec. ( 1000 x .01 = 1 x 10 = 10 )

1000 kg

1 kg

.01 m/sec

10 m/sec

Impulse and Momentum

• If momentum changes, it’s because mass or velocity change.
• Most often mass doesn’t change so velocity changes and that is acceleration.
• And mass x acceleration = force
• Applying a force over a time interval to an object changes the momentum
• Force x time = Impulse
• Impulse = Ft or Ft = mass x velocity

Ft = mv

MOMENTUM

• An object at rest has no momentum, why?
• Because anything times zero is zero
• (the velocity component is zero for an object at rest)
• To INCREASE MOMENTUM, apply the greatest force possible for as long as possible.
• Examples :
• pulling a sling shot
• drawing an arrow in a bow all the way back
• a long cannon for maximum range
• hitting a golf ball or a baseball. (follow through is important for these !)

FORCE

TIME

MOMENTUM

SOME VOCABULARY

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• impulse: impact force X time (newton^.sec) Ft = impulse

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• impact: the force acting on an object (N) usually when it hits something.

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• impact forces: average force of impact

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• Decreasing Momentum
• Which would it be more safe to hit in a car ?

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• Knowing the physics helps us understand why hitting a soft object is better than hitting a hard one.

MOMENTUM

mv

mv

Ft

Ft

MOMENTUM

• In each case, the momentum is decreased by the same amount or impulse (force x time)

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• Hitting the haystack extends the impact time (the time in which the momentum is brought to zero).

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• The longer impact time reduces the force of impact and decreases the deceleration.

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• Whenever it is desired to decrease the force of impact, extend the time of impact !

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DECREASING MOMENTUM

• If the time of impact is increased by 100 times (say from .01 sec to 1 sec), then the force of impact is reduced by 100 times (say to something survivable).

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• EXAMPLES :
• Padded dashboards on cars
• Airbags in cars or safety nets in circuses
• Moving your hand backward as you catch a fast-moving ball with your bare hand or a boxer moving with a punch.
• Flexing your knees when jumping from a higher place to the ground or elastic cords for bungee jumping
• Using wrestling mats instead of hardwood floors.
• Dropping a glass dish onto a carpet instead of a sidewalk.

EXAMPLES OF DECREASING MOMENTUM

• Bruiser Bruno on boxing …

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• Increased impact time reduces force of impact
• Bungee Jumping …

Ft = change in

momentum

Ft = change in

momentum

Ft = Δmv applies here.

mv = the momentum gained before the cord begins to stretch that we wish to change.

Ft = the impulse the cord supplies to reduce the momentum to zero.

Because the rubber cord stretches for a long time the average force on the jumper is small.

Questions :

• When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard ceramic tile floor ?
• The impulse would be the same for either surface because there is the same momentum change for each. It is the force that is less for the impulse on the carpet because of the greater time of momentum change. There is a difference between impulse and impact.

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• If a boxer is able to increase the impact time by 5 times by “riding” with a punch, by how much will the force of impact be reduced?
• Since the time of impact increases by 5 times, the force of impact will be reduced by 5 times.

BOUNCING

• IMPULSES ARE GREATER WHEN AN OBJECT BOUNCES
• The impulse required to bring an object to a stop and then to throw it back upward again is greater than the impulse required to merely bring the object to a stop.
• When a martial artist breaks boards,
• does their hand bounce?
• Is impulse or momentum greater ?

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• Example :
• The Pelton Wheel.

CONSERVATION OF MOMENTUM

• To accelerate an object, a force must be applied.
• The force or impulse on the object must come from outside the object.

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• EXAMPLES : The air in a basketball, sitting in a car and pushing on the dashboard or sitting in a boat and blowing on the sail don’t create movement.
• Internal forces like these are balanced and cancel each other.
• If no outside force is present, no change in momentum is possible.

The Law of Conservation of Momentum

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• Unless there is an external force acting on a system, the momentum of the system remains unchanged.

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• This means that, when all of the forces are internal (for EXAMPLE: the nucleus of an atom undergoing radioactive decay, cars colliding, or stars exploding the net momentum of the system before and after the event is the same.

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QUESTIONS

• 1. Newton’s second law states that if no net force is exerted on a system, no acceleration occurs. Does it follow that no change in momentum occurs?
• No acceleration means that no change occurs in velocity and therefore no change in momentum.

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• 2. Newton’s 3^rd law states that the forces exerted on a cannon and cannonball are equal and opposite. Does it follow that the impulse exerted on the cannon and cannonball are also equal and opposite?
• Since the time interval and forces are equal and opposite, the impulses (F x t) are also equal and opposite.

The Law of Conservation of Momentum

• No change in momentum occurs unless outside force acts
• If objects collide,

Total momentum _before = total momentum _after for both objects

• For any system, this holds true

COLLISIONS

• ELASTIC COLLISIONS

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• INELASTIC COLLISIONS

Momentum transfer from one

Object to another .

Is a Newton’s cradle like the one

Pictured here, an example of an

elastic or inelastic collision?

Problem Solving #1�(write this down)

• A 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is at rest. Find the velocity of the fish immediately after “lunch”.
• System is both fish, so …..
• net momentum _before = net momentum _after
• ((mv _{big fish}) + (mv_{small fish}))_before = (m _{big fish}+ m_{small fish})(V_both)after

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• (6 kg)(1 m/sec) + (2 kg)(0 m/sec) = (6 kg + 2 kg)(v_after)
• 6 kg^.m/sec = (8 kg)(v_after)
• v_after = 6 kg^.m/sec / 8 kg
• 8 kg
• v_after = ¾ m/sec

v_after =

Problem Solving #2

• Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 2 m/sec. Find the velocity of the fish immediately after “lunch”.
• System is both fish, so….
• total momentum _before = total momentum _after
• (total mv)_before = (total mv)_after
• ((mv _{big fish}) + (mv_{small fish}))_before = (mv _{big fish}) + (mv_{small fish}))_after

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• (6 kg)(1 m/s) + (2 kg)(-2 m/s) = (6 kg + 2 kg)(v_after)
• 6 kg^.m/sec + -4 kg^.m/sec = (8 kg)(v_after)
• v_after = 2 kg^.m/sec / 8 kg
• 8 kg
• v_after = ¼ m/sec

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v_after =

Problem Solving Quiz

1. Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 3 m/sec. Find the velocity of the fish immediately after “lunch”.

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2. Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 4 m/sec. Find the velocity of the fish immediately after “lunch”.

Problem Solving #3 & #4

• Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 3 m/sec.
• (net mv)_before = (net mv)_after
• (6 kg)(1 m/sec) + (2 kg)(-3 m/sec) = (6 kg + 2 kg)(v_after)
• 6 kg^.m/sec + -6 kg^.m/sec = (8 kg)(v_after)
• v_after = 0 m/sec

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• Now the 6 kg fish swimming at 1 m/sec swallows a 2 kg fish that is swimming towards it at 4 m/sec.
• (net mv)_before = (net mv)_after
• (6 kg)(1 m/sec) + (2 kg)(-4 m/sec) = (6 kg + 2 kg)(v_after)
• 6 kg^.m/sec + -8 kg^.m/sec = (8 kg)(v_after)
• v_after = -1/4 m/sec

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MOMENTUM VECTORS

• Momentum can be analyzed by using vectors
• The momentum of a car accident is equal to the vector sum of the momentum of each car A & B before the collision.

A

B

MOMENTUM VECTORS (Continued)

• When a firecracker bursts, the vector sum of the momenta of its fragments add up to the momentum of the firecracker just before it exploded.

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• The same goes for subatomic elementary particles. The tracks they leave help to determine their relative mass and type.