Kinematics is the study of motion without considering its causes
How does an object move?
Not why an object is moving?
2-1 Position, Displacement, and Distance
Position: The position of an object describes where the object is relative to a reference frame
Reference frame means another object that we can use for comparison.
Some reference frames can be stationary, while other reference frames can be moving.
Control the position of the block by moving the sliders below:
In the example above, the reference frame we are using is the x-y plane, with the bottom-right corner being the (0,0) point
Since we need two numbers to represent where the object is, this is a 2-dimensional reference frame
Displacement: When an object moves from one position to another,
its change in position is called displacement.
Δx = xf − x0
Displacement = final position − initial position
Greek letter delta (Δ) means "change in" whatever quantity follows it.
Distance: Distance is defined to be the magnitude (or size) of the displacement between two points
Note that the distance between two points does not depend on the direction
Unlike displacement, the distance between two points will never be negative.
Move the slider to change the position of the ball
Note the difference between displacement and distance.
Distance Traveled: The distance traveled is the total length of the path traveled between two positions.
2-2 Vectors, Scalars, and Coordinate Systems
Vectors: A vector is any quantity that has both a magnitude (size) and a direction
Some examples of vectors are:
Position
Displacement
Velocity (described below)
Vectors can be positive (+) or negative (−).
Vectors that point in opposite directions have opposite signs
Scalars: A scalar is any quantity that has a magnitude (size), but no direction.
Some examples of scalars are:
Distance
Time
Mass
Speed
Scalars are always positive numbers
Coordinate Systems: In order to describe the direction of a vector, we must first dsignate a coordinate system .
One-dimensional: For one-dimensional motion, our coordinate system consists of a one-dimensional coordinate line
When describing horizontal motion, we usually describe motion to the right as positive and motion to the left as negative
When describing vertical motion, we usually describe the up direction as positive and the down direction as negative
However, it can sometimes be more convenient to switch the positive and negative directions
For example, if an object is moving towards the left, it makes more sense to call that the +x direction, this is perfectly acceptable.
Two-dimensional:
For two-dimensional motion, our coordinate consists of the x-y plane
Again, we are free to switch the positive (+) and negative (−) directions (for both x and y).
2-3 Time, Velocity, and Speed
Time: Time is one of the most fundamental physical quantities. These quantities are defined by how they are measured.
Every measurement of time involves measuring a change in some quantity: a heartbeat, the position of the Sun in the sky, the position of Earth around the Sun, etc.
Thus we can say that time is change, or the interval over which change occurs
The SI unit for time is the second, abbreviated s
The time it takes for something to occur is called its elapsed time
Elapsed time is measured as:
Δt = tf − t0
(Notice the similarity between this formula and the one for displacement?)
For simplicity's sake, we will always say that t0 = 0, thus Δt = tf
Velocity: The velocity of an object is described as the rate of change of its position.
This means that velocity is a measurement of how fast the position of an object is changing.
In other words, velocity describes how fast an object is moving.
The average velocity of an object over a certain period of time is calculated as its total displacement divided by the total elapsed time.
v = Δx ⁄ Δ t
If the velocity of an object is zero, that means that the object is not moving (this is often called being at rest).
The instantaneous velocity of an object is the velocity of the object at a given moment in time.
Velocity is a vector, therefore it always has a magnitude and a direction
Thus velocity can be positive or negative
Press play to see how objects with different velocities move differently
Notice that the velocity becomes negative when an object is moving to the left, but its speed is always positive
Speed: The speed of an object is the magnitude of its velocity.
In everyday language, most people use the terms "speed" and "velocity" interchangeably. But in physics, they do not have the same meaning.
While velocity is a vector, speed is a scalar.
The average speed of an object over a given period of time is its total distance travelled divided by the elapsed time. (Note that this is different than the average velocity)
The instantaneous speed of an object at any given moment is the magnitude of its instantaneous velocity at that moment.
2-4 Acceleration
Acceleration: The acceleration of an object is the rate of change of its velocity
Acceleration is a measurement of how fast the velocity of an object is changinng
a = Δv ⁄ Δ t
Press play to see how objects with different accelerations move
Both of these objects start from rest, but the red square has a bigger accceleration than the blue square.
Even when the blue square has a head start
the red square catches up because it has a higher acceleration. Notice the blue square's initial velocity and position
Acceleration is to velocity what velocity is to displacement.
If the acceleration of an object is zero, that means the object is moving with a constant velocity
Acceleration is a vector and therefore has a magnitude and a direction
This also means that acceleration can be positive or negative
A negative acceleration means that the object is slowing down
This can sometimes be called deceleration
But deceleration is not always the same as negative acceleration
Press play to see the difference between deceleration and negative acceleration
The red square is decelerating
The blue square has a negative acceleration
2-5 Motion Equations for Constant Acceleration in One Dimension
Different equations are used for finding an object's properties depening on whether its velocity is constant or not.
Assume acceleration is constant (a = a = constant).
These equations do not require acceleration in their calculations:
v = Δx ⁄ Δ t = (x-x0) ⁄ t = (v+v0) ⁄ 2
x = x0 + vt
Solving for final velocity requires the use of a constant acceleration:
a = Δv ⁄ Δ t = (v - v0) ⁄ t
v = v0 + at
Solving for final position also requires a constant acceleration:
x = x0 + v0t + 1 ⁄ 2 * at2
Final velocity can be solved without the use of a time variable:
v2 = v02 + 2a(x - x0)
2-6 Problem-Solving Basics for One-Dimensional Kinematics
In order to understand what is being asked for in a problem and figure out how to solve it, the following steps can be used to simplify the process:
Step 1 - Analyze the problem to see which physics concepts are used.
Step 2 - Identify given information, such as values.
Step 3 - Idenfity the unknown information that is being asked for.
Step 4 - Find an equation (or equations) that may lead you to the desired answer (the unknown information).
Step 5 - Input the given information into the equation(s) and calculate the solution (with units).
Step 6 - Analyze the answer to make sure it is reasonable.
2-7 Falling Objects
If air resistance and friction are ignored, then all objects in a certain location on the earth fall to the ground at the same constant acceleration, no matter their mass.
Air resistance is what causes lighter objects to fall slower than heavy objects. If an object falls without air resistance or friction, then it is in free-fall
Gravity acts on all objects on Earth, causing them to fall towards the ground at a constant acceleration.
Assuming upward direction is positive and gravity is g:
a = -g = -9.80 m/s2
Objects that only move vertically (such as those in free-fall) only have a vertical velocity. Ignoring air resistance and friction, these objects only move in one dimension with constant acceleration g.
