Momentum and energy
Isaac Newton formulated three laws about motion:
- The Law of Inertia
- F = ma
- For every force there is an equal and opposite force
Newton's First Law
The Law of Inertia
Every object continues in a state of rest or of uniform speed in a straight line unless acted upon by a nonzero force.
You may have heard this as
An object in (constant) motion stays in (constant) motion, and object at rest stays at rest.
Newton's First Law
The key to this idea is "continues" -- an object "continues" doing what it is already doing (in that direction).
Think of the plates and silverware in the tablecloth demonstration: they are at rest and remain at rest when the tablecloth is quickly pulled away.
You have to be careful, though, because sometimes the idea can get tricky:
newton's first law
Examples of Newton's First Law
- When you slam on the brakes in your car, your body keeps going forward (one good reason to wear your seat belt).
- When you are rear-ended in your car, your head stays where it is, giving you whiplash if you don't have a headrest (this is sometimes tricky -- why does your head seem like it is going backwards? Is it really "going backwards"?)
- The Voyager I spacecraft, launched in 1977, is traveling away from Earth at 38,600mi/hr, but it hasn't had propellant since 1978.
- Flip a coin while on an airplane -- the coin doesn't fly backwards at 500mi/hr -- it lands in your hand.
newton's second law
F = ma
a = ------
This formula says that the acceleration of an object is directly proportional to the net force on an object, and inversely proportional to the mass of the object.
In other words, push harder, get more acceleration. Make bigger, get less acceleration.
finally! Why objects fall at the same rate
Hammer and Feather Video:
In a vacuum, a hammer and a feather fall equally, side by side. Would it be correct to say that equal forces of gravity act on both the hammer and the feather in a vacuum?
Falling with Air Resistance
(also: cats and terminal velocity)
Newton's Third Law
We've been discussing forces as "pushes" and "pulls." The fact is that a force cannot act on its own: forces always come in pairs in an interaction.
- Support yourself on a wall. Your hand pushes on the wall, and the wall pushes back on your hand (and you can feel it)
- A sledge-hammer strikes a stake: the hammer pushes on the stake, and the stake pushes back on the hammer (and the hammer stops)
- You pull on a rope attached to a sled. The sled pulls back on you (and you feel it).
- A boxer punches a wall (wall pushes back) / a boxer punches a piece of paper (paper pushes back, but overall push isn't as much--neither is the punch!)
newton's third law
Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first.
If we say that there is an "action force" and a "reaction force," then we can say:
To every action there is always an opposed equal reaction.
(by the way, it doesn't matter which force is the action and which force is the reaction)
newton's third law
ACTION: Object A exerts a force on object B.
REACTION: Object B exerts a force on object A.
- Hammer pushes down on nail : nail pushes up on hammer.
- Foot kicks ball forward : ball pushes backwards on foot.
- Book pushes down on table : table pushes up on book.
- Earth pulls down on human through gravity : ???
newton's third law
Identify the force pair when a rifle shoots a bullet.
If the forces are equal and opposite, why does the bullet travel forward at hundreds of miles per hour, while the gun recoils backward at a much slower speed? It's all about acceleration:
Defining the System
If action and reaction forces are equal and opposite, why don't they cancel each other out?
defining the system
If the system is both the orange and the apple, they will cancel, so we need to consider what happens outside that system.
Why can't you push your car when it breaks down by pushing on the dashboard?
It is harder to stop a giant truck moving at some speed than it is to stop a small car moving at the same speed.
We say that the truck has more momentum than the car. By "momentum," we mean "inertia in motion." Specifically,
Momentum = mass X velocity = mv = p
Why "p"? Nobody remembers (but "m" was already taken)
Can a car ever have more momentum than a truck?
Momentum and Impulse
What happens when momentum changes?
Mass generally stays the same (although a piece could fall off, or be added to an object).
If the velocity changes, this is just an acceleration. What produces accelerations? Forces. The greater the force, the greater the acceleration.
But, there is another aspect: the amount of time the force acts. If a force acts for a longer time, there is more of an acceleration, and more of a change in momentum.
momentum and impulse
Impulse = force X time = Ft = Change in Momentum
In other words:
impulse = Ft =Δ(mv)
We can derive this from our a=F/m definition and from a=Δv/Δt
We call "Ft = Δ(mv)" the "impulse-momentum relationship"
the impulse–momentum relationship
Case 1: Increasing Momentum
Sports: Apply the greatest force for as long as possible
E.g., baseball, golf ball, drag racing
Case 2: Decreasing Momentum Over a Long Time
Apply a small force over a long time
E.g., stopping an out of control truck: wall or long sand pit? (it's the same impulse!)
Air Bags: the time is relatively short, but longer than hitting the steering wheel or dashboard
Acrobatic Safety net: "cushion effect" by increasing stop time
Case 3: What about Bouncing?
conservation of momentum
In a system, total momentum is always conserved.
Firing a cannon: Initially, the momentum is zero.
Upon firing, the cannonball travels in one direction with a momentum, p, and the cannon itself travels in the opposite direction with a momentum -p. Momentum is a vector quality, so it can cancel!
When a pool ball hits another pool ball head-on, the first ball stops, and the second ball gains the momentum from the first. Overall, the total momentum is the same.
The Law of Conservation of momentum
In the absence of an external force, the momentum of a system remains unchanged.
Look at examples of elastic and inelastic collisions (interactive)
An elastic collision is a collision where there is no lasting deformation or generation of heat. Elastic collisions conserve momentum and energy.
An inelastic collision is a collision with lasting deformation and/or the generation of heat. Completely inelastic collisions result in two bodies fusing as one.
energy and work
How much faster will you hit the ground if you fall from twice the height?
(we'll get to that by the end of the chapter!)
Energy is the property of a system that enables it to do work.
(this definition works, but it is not perfect!)
Work is defined as 'force X distance'
We have to be very careful about what we mean by "distance"
When you push a crate across a floor, you do work, but you don't do work when you carry a book at constant speed across the room.
energy and work
W = F x d
If you lift two boxes up one story in a house, you do twice as much work as if you lifted one box. This is because the Force is twice as much.
If you lift one box two stories, it is twice as much work as if you lift one box one story. This is because the distance is twice as much.
How much work do you do if you hold a heavy weight above your head?
work and energy
How much work do you do carrying a heavy weight at constant velocity around the room?
There is a big difference between the work done on an object to the work done to hold an object (that would be internal to your muscles).
In some cases, an object may "store" energy because of it's position. Examples include: an object in a gravitational field (like on Earth); a rubber band that has been stretched; a spring that has been compressed.
Gravitational Potential Energy (PE) is the energy due to an elevated position.
Because the upward force needed to raise an object at constant speed is the object's weight (mg), we can define PE as:
PE = mgh (Measured in "Joules")
Because potential energy only depends on the height, it does not matter how the weight gets to that height!
The work done elevating the weight is the same in each case.
If an object is moving, it is capable of doing work.
We say that the object has kinetic energy (KE), and it depends on the object's mass and it's speed.
KE =½mv² (also measured in Joules)
If you throw a ball, your hand does work on the ball to give it kinetic energy. The ball can now do work on something when it hits it.
The kinetic energy is equal to the amount of work required to bring it to rest:
Net force X Distance = kinetic energy
Notice that the velocity is squared in the kinetic energy equation. This means that if an object is twice as fast, it can do four times as much work. If it is ten times as fast, it can do 100 times as much work!
This is one reason it takes your car four times as far to stop if you are going twice as fast!
conservation of energy
Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount never changes.
Interactive Pendulum demo
A pendulum bob conserves energy as it goes back and forth
Energy Conservation Calculation Video
conservation of energy
1. How much total energy does the cart have?
1. How fast is the roller coaster traveling initially? hint: v = sqrt(2KE/m)
2. How fast is the roller coaster traveling at the bottom of the hill?
3. How fast is the roller coaster traveling at the top of the loop?
4. How fast is the roller coaster traveling at the top of the final hill?
5. How high would the final hill have to be to stop the coaster?
Conservation Of Energy
Bowling Ball Video
Also, MIT professor Walter Lewin:
Why is it harder to run up a hill than to walk up a hill?
The amount of work is the same whether you walk or run!
Power is how fast work is done.
Power = -------------------
Power is measured in Watts
Wat's a Watt?
A watt is a Joule per second:
1 J / s = 1 W
(we used to measure power in "horsepower", 1hp = 746W)
What unit for power was used in Back to the Future?
Sources of Energy
Where does the energy to power our AC units come from?
Check questions for chapters
2 and 3