Finally, four-dimensional space includes time along with the three spatial coordinates ( $x, y, z, t$ ). This allows us to describe how an object changes over time.

SI units for physics measurements organized by base quantities

Vectors, components

A scalar is a quantity defined only by its magnitude (size), with no direction - such as length, time, or mass.

A vector has both magnitude and direction - for example, displacement, acceleration, or force.

The components of a vector are its projections along chosen axes. In other words, components break the vector into parts that tell you how much of the vector lies along each axis. Each component is a number (an amount), so components are scalars.

A common mnemonic for remembering trigonometric ratios is SOH CAH TOA. Some people like to sound it out (“Sock a Toe-a”), while others prefer a phrase, such as:

Some Old Hippie Caught Another Hippie Tripping on Acid

Either way, it helps you remember:

sin $θ$ is the ratio of the opposite side to the hypotenuse
cos $θ$ is the ratio of the adjacent side to the hypotenuse,
tan $θ$ is the ratio of the opposite side to the adjacent side.

Vector addition

You can add vectors directly only when they share the same direction.

When vectors point in different directions, decompose each vector into its $x, y$ , and $z$ components, then add the components separately to form the resultant vector. The idea is that if you add all the components back together, you reproduce the original vector.

An operation between two vectors may yield either a vector or a scalar. For example, squaring a vector’s magnitude to calculate kinetic energy produces a scalar.

In contrast:

Combining a vector with a scalar always results in a vector.
Operations between two scalars always yield a scalar.

Sample:

Add vectors $C D$ and $D E$ , where $C D = (3, 4)$ and $D E = (2, 6)$

(spoiler)

$C D + D E = (3, 4) + (2, 6)$ $\Rightarrow C D + D E = (3 + 2, 4 + 6)$ $\Rightarrow C D + D E = (5, 10)$

Speed, velocity (average and instantaneous)

Speed is a scalar that measures how fast an object moves. It describes the rate at which distance changes, without considering direction.

Velocity is a vector. It describes the rate at which displacement changes, and it includes direction.

Average speed is total distance divided by elapsed time. Average velocity is net displacement divided by the time interval.

Instantaneous speed is the speed at a specific moment (over an infinitesimally small time interval). It equals the magnitude of instantaneous velocity. Unlike instantaneous speed, instantaneous velocity includes direction, and that direction is tangent to the path at that point.

Average speed

$V = \frac{d}{t}$

$V_{a vg} = \frac{d}{t}$

Average velocity

$V = \frac{s}{t}$

$V_{a vg} = \frac{s}{t}$

Average acceleration

$a = \frac{Δ V}{Δ t}$

$a_{a vg} = \frac{V _{f} - V _{i}}{t}$

( $Δ V = V_{f} - V_{i}$ , i.e., final velocity minus initial velocity)

Acceleration

Average acceleration is the rate at which velocity changes over time (final velocity minus initial velocity).

In uniformly accelerated motion along a straight line, acceleration is constant and the direction does not change. In that special case, the numerical values of speed and velocity, and of distance and displacement, can be treated as the same - as long as you still track direction with your sign convention.

Under these conditions, the standard equations of uniform acceleration apply:

$s = V_{a vg} \cdot t$

$V_{a vg} = \frac{V _{f} + V _{i}}{2}$

$V_{f}^{2} = V_{i}^{2} + 2 a s$

$s = \frac{1}{2} a t^{2} + V_{i} t$

$d_{f} = d_{i} + s or l_{f} = l_{i} + s$

( $d$ = distance or $l$ = length, take your pick)

Refresher about formulas, which you should both memorize and be able to manipulate:

Assign a consistent sign convention by designating one direction as positive and the opposite as negative, then use that convention consistently in every calculation.

In Cartesian (graphing) coordinates, upward and rightward movements are typically positive, while downward and leftward are negative.

For free falls, however, it’s common to define downward as positive (falling is what’s expected, so think of it as gain/progress).

Freely falling objects move under constant acceleration due to gravity, which on Earth is approximately $g$ = $9.8 m / s^{2}$ . Whether an object is dropped, thrown upward, or projected at an angle, it is considered to be in free fall. For ease of calculation, it’s common to assign downward as positive so that $g$ remains positive for objects dropped or thrown downward; when an object is tossed upward, its initial velocity has the opposite sign to $g$ .

In ideal conditions without air resistance, the net acceleration remains constant at $g$ because the force of gravity (weight) and the mass of the object are constant. When air resistance is present, the acceleration decreases over time as resistive forces build up, until eventually reaching terminal velocity, where the gravitational force is balanced by air resistance and acceleration ceases.

Projectiles are a specific case of free-falling bodies. Their vertical motion is always accelerated downward at $g$ , while their horizontal motion remains constant due to the absence of horizontal forces. To compute the time a projectile remains in the air, only the vertical component is considered; the horizontal distance traveled is then the product of this time and the constant horizontal velocity.

When an object is tossed straight upward and returns to its starting point, the overall displacement is zero because the upward and downward movements cancel each other out. In such cases, the time spent ascending equals the time spent descending.

Orbiting satellites are also in a state of free fall, continuously accelerating toward the Earth due to gravity. However, because they have sufficient tangential velocity, the Earth’s surface curves away at the same rate, preventing the satellite from colliding with the planet.

This consistent application of uniform acceleration in free fall underlies many calculations in physics, where the distinction between instantaneous and average values is crucial, and the direction of motion is accounted for through a chosen sign convention.

Displacement

Displacement is a vector quantity that represents the change in an object’s position. It is defined by both its magnitude (the shortest distance between the starting point and the endpoint) and its direction.

Unlike distance, which is a scalar that measures the total path length traveled, displacement focuses solely on the net change in position, regardless of the path taken.

Average speed versus velocity during a round trip

Graphing position

These simple models show how velocity, speed, and position can be graphed as functions of time. As motion includes more changes in speed or direction, the graphs become more complex.

Position, velocity, and speed graphs on a trip

Units and dimensions

1D: single length measure; 2D: x-y plane (length, width); 3D: x, y, z (adds depth/volume); 4D: x, y, z, t (includes time)
SI base units: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd)
SI prefixes: kilo-, centi-, milli-, micro-, etc. for scaling units

Vectors, components

Scalar: magnitude only (length, time, mass)
Vector: magnitude and direction (displacement, force, acceleration)
Components: projections along axes; each is a scalar representing vector’s effect in that direction

Vector addition

Add directly only if vectors share direction
For different directions: break into x, y, z components, add components separately
Vector + vector: result can be vector or scalar; vector + scalar: always vector; scalar + scalar: scalar

Speed, velocity (average and instantaneous)

Speed: scalar, rate of distance change ( $V = \frac{d}{t}$ )
Velocity: vector, rate of displacement change ( $V = \frac{s}{t}$ )
Average: total distance/displacement over time; Instantaneous: value at a specific moment

Acceleration

Average acceleration: change in velocity over time ( $a = \frac{Δ V}{Δ t}$ )
Uniform acceleration: constant acceleration, direction unchanged; use kinematic equations
- $s = V_{a vg} \cdot t$ , $V_{a vg} = \frac{V _{f} + V _{i}}{2}$ , $V_{f}^{2} = V_{i}^{2} + 2 a s$ , $s = \frac{1}{2} a t^{2} + V_{i} t$
Free fall: $g = 9.8 m / s^{2}$ (downward positive by convention), applies to dropped, thrown, or orbiting objects

Displacement

Vector: change in position (magnitude + direction)
Displacement ≠ distance (distance = total path, displacement = net change)

Graphing position

Position, velocity, and speed can be graphed vs. time
More complex motion = more complex graphs; velocity is slope of position-time graph

Translational motion

Units and dimensions

A one-dimensional space involves only a single measure of length or distance.

In a two-dimensional framework, objects are defined on a flat plane using an x-y coordinate system, which includes both length and width.

Three-dimensional space adds depth. Objects are described by x, y, and z coordinates, which lets us represent volume.

Finally, four-dimensional space includes time along with the three spatial coordinates ( $x, y, z, t$ ). This allows us to describe how an object changes over time.