Because any translational motion problem can be separated into one or more 1-dimensional problems.

Problems are often analyzed this way - a complex problem can often be reduced to a set of simpler problems.

Scalars and vectors

A scalar is something that has only a magnitude while a vector has both a magnitude and a direction. In 1-dimension it's hard to tell them apart!

• Always draw a diagram of what you're trying to analyze.
  
  
• Choose an appropriate coordinate system. Define an origin (a reference point) and choose positive and negative directions.
  
  
• To fully describe motion you need to account for four things:
  time (scalar)
  position (vector)
  velocity (vector)
  acceleration (vector)

Displacement is a vector representing the distance traveled and specifying the direction.

If you start at position x_o and move to position x, your displacement Δx is defined as:

Example

If you move 5 meters north, Δx = 5 m north.

Now go the other direction, with a displacement of 3 m south.

The total distance traveled is 8 m. What is your net displacement?

Δx₁ = +5 m north
Δx₂ = +3 m south = - 3 m north

Net displacement: Δx = Δx₁ + Δx₂ = +5 -3 = +2 m north

• Velocity is a vector representing how fast, and in what direction, something is traveling.
• Speed is a scalar. It is the magnitude of the velocity.
• Speed just represents how fast something is going.

Velocity is the rate of change of position

Average velocity:   vav

=

Δx Δt

=

xf - xi tf - ti

Instantaneous velocity: v

=

dx dt

The instantaneous velocity is the slope of the line on a position-versus-time graph.

If the velocity is constant the instantaneous velocity is equal to the average velocity.

Turning the last equation around, the displacement can be found from:

Displacement:   Δx

=

∫

v(t) dt

• Acceleration is a vector representing how fast, and in what direction, something's velocity is changing.
• Acceleration is related to velocity the same way velocity is related to position.

Acceleration is the rate of change of velocity.

Average acceleration:   aav

=

Δv Δt

=

vf - vi tf - ti

Instantaneous acceleration: a

=

dv dt

The instantaneous acceleration is the slope of the line on a velocity-versus-time graph.

If the acceleration is constant the instantaneous acceleration is equal to the average acceleration.

Turning the last equation around:

Change in velocity:   Δv

=

∫

a(t) dt

These equations relate displacement, velocity, acceleration, and time, and apply under the following conditions:

• the acceleration is constant
• the motion is measured from t = 0
• the fact that several variables are vectors is accounted for with appropriate plus and minus signs

The equations are derived from the definitions of acceleration, velocity, and displacement.