Momentum and Newton’s 2nd Law of Motion

Everyday Language

What do you say when a sports team is on a roll?

They may not have the lead but they may have …..

MOMENTUM

• A team that has momentum is hard to stop

Momentum

• Momentum is defined as “inertia in motion” or “mass in motion”

• It carries the notion of both mass and velocity –Velocity is defined as change in

distance over a period of time • Speed with a direction

Momentum

• The momentum of an object is the product of its mass and velocity Momentum = mass x velocity

p=mv

p = (kg) x 𝑚

𝑠

Momentum

• In symbols:

p = mv p

m v

Equivalent Momenta

Bus: m = 9000 kg; v = 16 m /s

p = 1.44 ·105 kg · m /s

Train: m = 3.6 ·104 kg; v = 4 m /s

p = 1.44 ·105 kg · m /s

Car: m = 1800 kg; v = 80 m /s

p = 1.44 ·105 kg · m /s

continued on next slide

Equivalent Momenta (cont.)

The train, bus, and car all have different masses and

speeds, but their momenta are the same in

magnitude. The massive train has a slow speed; the

low-mass car has a great speed; and the bus has

moderate mass and speed. Note: We can only say

that the magnitudes of their momenta are equal since

they’re aren’t moving in the same direction.

The difficulty in bringing each vehicle to rest--in terms

of a combination of the force and time required--

would be the same, since they each have the same

momentum.

How hard is it to stop a moving object?

• In order to stop an object, we have to apply a force over a period of time

• This is called an impulse

Impulse = FΔt

Units: N∙s

F = force (N) Δt = time elapsed (s)

Impulse – Momentum Theorem

vmFt

IMPULSE CHANGE IN MOMENTUM

This theorem reveals some

interesting relationships such

as the INVERSE relationship

between FORCE and TIME

t

vmF

Imagine a car hitting a wall and coming to rest. The force on the car due to the wall is

large (big F ), but that force only acts for a small amount of time (little t ). Now imagine

the same car moving at the same speed but this time hitting a giant haystack and

coming to rest. The force on the car is much smaller now (little F ), but it acts for a

much longer time (big t ). In each case the impulse involved is the same since the

change in momentum of the car is the same. Any net force, no matter how small, can

bring an object to rest if it has enough time. A pole vaulter can fall from a great height

without getting hurt because the mat applies a smaller force over a longer period of time

than the ground alone would.

Stopping Time

F t = F t

Impulse – Momentum Relationships

Impulse Changes Momentum

A greater impulse exerted on an object A greater change in momentum

OR

Impulse = Change in momentum

OR

Impulse = Δ(mv) Greek symbol “Delta”

Means “the change in…”

• The impulse exerted on an object equals the object’s change in

momentum

• Impulse can be exerted on an object to either INCREASE or DECREASE its

momentum.

Impulse & Momentum

• In symbols:

I = p

Newton’s Second Law of Motion

Recall That… • To summarize this unit so far:

– Newton’s 1st Law of Motion states that an object will “keep doing what it’s doing” unless acted upon by an external force

– In order to change one’s momentum, a force must be exerted on an object over a period of time

– Or the change in momentum of a system is equal to the net impulse acting on the system

• So according to this statement, force is defined as a change in

momentum with a change in time

F = ∆(mv)

∆t

Momentum & Newton’s 2nd Law of Motion

• Since mass is essentially constant for an object, we can derive Newton’s 2nd Law of Motion:

F = m∆(v)

∆t

• This equation shows that a force causes a change in velocity – And likewise, a change in velocity generates a

force

SUPPORTS NEWTON’S 1ST LAW OF MOTION!

Deriving Newton’s 2nd Law of Motion

• The change in velocity divided by the change in time (

∆(v)

∆t ) is the definition of acceleration, a

– Acceleration is more generally defined as when the motion of an object changes speed or direction • Speed up • Slow down • Changes direction

• So, Newton’s 2nd Law of Motion becomes the more familiar:

F = ma

Therefore: Force and Mass

Determine Acceleration!

F = ma

**WORKSHEET WITH EXAMPLES AND DESCRIPTIONS OR ACCELERATION**

Newton’s 2nd Law • Newton’s Second Law states that an object

acted upon by an unbalanced force will accelerate in the direction of the force

• If you kick the ball, it starts moving

• The ball accelerates only while your foot is in contact with the ball

Relation of Force to Acceleration

• a is the acceleration

• m is the mass,

• and F is the net force.

F = ma

Units for Force and Acceleration

Unit of Force = Newton (N)

Unit of mass is kg

Unit for acceleration is m/s2

What does F = ma Mean?

• This equation tells us that an object subjected to an external force will accelerate

• Furthermore, this equation tells us that the amount of acceleration is proportional to the size of the force

• The amount of acceleration is inversely proportional to mass of an object – For equal forces, a more massive object will experience less

acceleration than an object with a smaller mass

More about F = ma

• If you double the mass, you double the force

• If you double the acceleration, you double the force

• If you double the mass but keep the force the same, the acceleration will decrease by half

• What if you double the mass and the acceleration?

(2m)(2a) = 4F

– Doubling the mass and the acceleration quadruples the force

What does F = ma say?

Something very small (low mass) that’s changing speed very quickly (high acceleration), like a bullet, can still have a great force. Something very small changing speed very slowly will have a very weak force.

Something very massive (high mass) that’s changing speed very slowly (low acceleration), like a glacier, can still have great force.

Plug in Practice

A book with a mass of 2.0kg is pushed along a table. If the net force on the book is 1.0N, what is the book’s acceleration?

Answer: .5m/s2

Circular Motion

• A rider on a merry-go-round ride moves in a circle

• This type of motion is called circular motion • If you are in circular

motion, your direction of motion is constantly changing.

• This means you are constantly accelerating

Circular Motion

• If you are constantly accelerating, there must be a force acting on you the entire time

• The force exerted is the centripetal force and always points toward the center of the circle

Gravity & It’s Relation to Force

• Gravity is the force of attraction that exists between any two objects that have mass

• The force of gravity depends on the mass of the objects and the distance between them

Gravity & Weight

• The force of gravity causes all objects near Earth’s surface to fall with an acceleration of 9.8 m/s²

• Thus, all objects in free fall have an acceleration of 9.8 m/s²

• Your weight on Earth is the gravitational force between you and Earth

F = ma

F = Weight when a is due to gravity!

Gravity and Weight

• So, how are weight and mass different?

• Weight is a force, like the push of your hand is a force, and is measured in Newtons

• Mass is the amount of matter in an object, and doesn’t depend on location

• Weight will vary with location, but mass will remain constant!