Mechanics

Displacement — (change in position)
Displacement is the straight-line distance between a start and end point. It has a direction, and direction must always be specified.

Has magnitude! Magnitude is the value, or amount, without direction.

Direction
There are three main ways to describe direction:

Distance
Distance is a unit of measure — it is the integral of the change in position over time, I guess. Always greater than or equal to the magnitude of the displacement.

Velocity
Velocity is the direction and magnitude of distance per time:

Speed
Speed is the magnitude of velocity when the object moves in a straight line — displacement distance traveled!

This uses distance traveled and NOT displacement. Speed does not have direction.

Position as a Function of Time

So it's just calculus!

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Acceleration
Acceleration is the change in velocity with respect to time.

Acceleration is measured in (base units) meters per second per second (). Since it is in terms of velocity (which is in terms of displacement), acceleration has both magnitude and direction.

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Velocity as a Function of Time
Velocity, measured in , is on the vertical axis. Time, like in the position graph, is on the horizontal axis. Then, the slope of the graph is the change in velocity with respect to the change in time... which is acceleration! So it's just calculus.

Uniformly Accelerated Motion (UAM)
Surprisingly enough, an object in UAM is moving at a constant (uniform) acceleration. Some examples include a ball rolling down an incline, an object falling, or an object being pushed up by buoyant force.

UAM Equations

Equation must be true because obviously the final velocity is the sum of the initial velocity and the acceleration applied over a period of time. Equation is true because the change in position (with direction) is equal to how much the initial velocity changes the position plus the amount that the position is changed by due to acceleration (). Equation is true because . If we square both sides, we get . We know that is equal to times the displacement under constant acceleration and is the displacement under constant velocity. So, we get . Equation is obvious — the change in position (with direction) is equal to the average velocity over a period of time times the change in time. Cool stuff. But wait — we have APC to cover! APC equations:

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Free Fall
https://www.youtube.com/watch?v=PIuAFrLeXfY&t=62s
An object is in free fall if the only force acting on it is the force of gravity. Basically, it must not be touching any other object. And, of course (welcome to physics), there is no air resistance. On Earth (most of the time, ish), an object in free fall has acceleration (in the direction) equal to .

Mass is irrelevant!

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Projectile Motion
There are two components to projectile motion: and movement. In the direction, velocity is constant, so . In the direction, the object is in free fall. So, it follows the UAM equations.

Projectile Motion Range
The range of projectile motion is defined as the horizontal displacement () when the overall vertical displacement () is .

Relative Motion
Velocity is measured from a frame of reference — from the Earth's perspective, an object could be moving, but from its own perspective, that object is stationary and the Earth is moving.

A negative velocity of A relative to B is the same as the velocity of B relative to B. So, whenever we sum the velocity of A relative to B and the velocity of B relative to C, we get the velocity of A relative to C.

Inertia
The tendency of an object to resist a change in its state of motion. Basically, the tendency of an object to resist acceleration. An object in motion will stay in motion, and one at rest will stay at rest. Type shit.

Inertial Mass
A measure of inertia, or a measure of an object's resistance to acceleration. Experimentally identical to gravitational mass.

Force
The ability to change the state of motion of an object (apply acceleration). There are two types of forces:

Force of Gravity ()
The attraction that exists between the Earth and an object. , but we know that , so , where is gravitational mass. is DOWN (drop something). Note that the base SI dimension for is . For English units, pounds ().

Free Body Diagrams (FBD)/Force Diagram
The diagram of all the forces acting on a freed (singled-out) object. Noteworthy forces:

Center of Mass
The location at which we consider all of an object's mass to be concentrated.

Newton's Laws of Motion

Tension Force ( or )
Tension Force is the force transmitted through a rope, cable, string, or wire pulled taut by forces acting on both ends. Tension is always a pull, is always directed along the rope, and is always in opposite directions on both ends of the rope but is equal in magnitude.

Equilibrium
An object is in equilibrium if it has no net force acting on it. That is, it has zero acceleration. Types of equilibrium:

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Force of Friction
There are two types of friction:

Coefficient of Friction ()
The coefficient of friction defines the ratio of the force of friction to the normal force, meaning that it is dimensionless. There is no theoretical way to calculate it, and the best/only way of finding it is experimentally. The coefficient of static friction is always greater than the coefficient of kinetic friction: . A good range (for solution sanity checks) is , with some high-friction combinations going up to .

Center of Mass
The center of mass is the mass-weighted average position of mass in an object. At the center of mass of an object, a force applied will only cause linear acceleration — elsewhere, it would cause a rotation as well. The center of mass of a system of objects is like the center of mass of any other object. Equation for the center of mass of a system:

Center of Mass with Integration
If we consider an object to be a collection of infinitely many particles, we can use the formula we had above to get that the position of the center of mass () is
How do we take the integral of position with respect to mass? Typically, we can express mass as a density function. As a simple example, consider an object with uniform density . is the volumetric density, and is equal to, well, the density: . Here, is total mass and is total volume (not velocity!). But, we know that the density of the infinitesimally small particles is the same as the density of the overall object, so we can solve for :
But what's ? We can express in terms of (the representation will be different for every shape, unfortunately), most of the time. If you do the calculations, you'll find that the center of mass of a triangle with base length and height has -position center of mass at if it has a uniform density or thickness.

Density
Thought there was only one type of density? You thought wrong! The one you were probably thinking of was the one we just saw, volumetric mass density:
There "are three," but actually a lot more than three. The three that are used in AP Physics are

Point Particle
A point particle is an object whose size and shape are so small that they are considered irrelevant to the situation, and their mass can be assumed to be in a single point in space.

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Terminal Velocity
In reality, air is a thing (and free-fall is not), so we have to take into account the force of drag. We know that the force of drag is dependent on velocity, so there's a velocity at which the force of drag equals the force of gravity and the object experiences no acceleration. The equation of terminal velocity is

An object that is falling (only gravity and drag) where DOWN IS UP has the following equations hold true:

Time Constant ()
The time constant is what goes in the denominator (??) in the expression . Measured (base dimensions) in seconds. Why is it like that? Because this means that the time constant is the time it takes for an object (falling) to reach of its terminal velocity. In other words, every increase in time by means getting closer to the object's terminal velocity.

Work ()
The equation for work is (remember to only use the magnitudes of and ):

Net Work Equals Theorem (Work-Energy Theorem)
Net work (haha network) is the change in kinetic energy.

Spring Force
Hooke's Law states that the force of a spring is given by

Elastic Limit
The elastic limit is the maximum displacement before permanent deformation. Hooke's Law applies only up to the elastic limit.

Kinetic Energy
Kinetic energy is the energy associated with the movement of an object. Measured in Joules. Cannot be negative!

Gravitational Potential Energy ()
Gravitational potential energy (which has the potential to become different types of energy) is the energy stored in an object due to its elevation.

Elastic Potential Energy ()
Elastic potential energy is the energy stored in an object due to the temporary deformation of that object. Stretching and compressing a spring is an example of elastic potential energy.

Mechanical Energy
Mechanical energy is the sum of potential and kinetic energies in a system. The work due to friction is the change in mechanical energy (mechanical energy becomes heat) when the force applied is zero:

Conservation of Mechanical Energy
Energy is not created or destroyed — it only changes forms. Most of the time. It is conserved as long as no energy is converted to heat, light, or sound energy (it has to stay mechanical for mechanical energy to be conserved). Since friction causes energy to be converted to heat, work done by friction must be zero. Also, work by the force applied must be zero as well (lifting something introduces gravitational potential energy into the system). As an extension of this, when the work done by is zero, .

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Conservative Force
A conservative force is one where the work done on an object by the force is the same regardless of the path taken by the object.
Ex. Force of gravity, spring force, electromagnetic force, magnetic force.

With conservative forces,

Change in Energy
The change in energy of a system is equal to the sum of the energy transferred into or out of the system:

Energy can be transferred through

Momentum ()
Momentum is the product of mass and velocity of an object (so it has magnitude and direction). Its units are , which has no special name. You can think of it as Newtons per second, though.

Force of Impact
We can find a new equation for Newton's second law (net force is mass acceleration):

For a calculus-based course, we also use the second equation, marked .

Impulse ()
The change in momentum is called impulse. It's equal to the change in force times the change in time. The units are Newtons seconds. For APC,

Isolated System
A system is isolated if the net force acting on it is zero.

Conservation of Momentum
In a closed system (and all collisions and explosions), momentum is conserved:

More precisely, the sum of the final momenta of the system is equal to the sum of the initial momenta of the system.

Power
Power is work over change in time. So, power is the rate at which work is done.

This is measured in Joules per second, or watts. Check the units table at the very beginning of these notes for conversions. Note that the velocity in this equation is average — don't make the mistake of using final velocity unless you are asked to solve for instantaneous power. The equation marked with actually can't solve for instantaneous power — only for average. The second equation can solve for both: you just have to replace with . For APC, there's another way to calculate instantaneous power:

Here, power is the dot product of the force and the velocity. Actually, we can go further:

Power is the change in energy over the change in time because power is the rate at which energy is transferred. We can rearrange and take the integral to get something even cooler:

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Elastic Collision
In an elastic collision, the objects bounce off of each other. Kinetic energy and momentum are conserved.

Inelastic Collision
Kinetic energy is NOT conserved (momentum is). The object(s) deform, which causes the object(s) to heat up, and kinetic energy is transferred. There are also perfectly inelastic collisions, where the objects stick to each other. All real-world bounce collisions are inelastic collisions.

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