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

Instantaneous velocity: v_x = dx/dt

Average velocity = v_avg = Δx/Δt

Δx = ∫ v_x dt

On your way to class one morning, you leave home and walk at 3 m/s east towards campus. After exactly one minute you realize that you've left your physics assignment at home, so you turn around and run, at 6 m/s, back to get it. You're running twice as fast as you walked, so it takes half as long (30 seconds) to get home again.

Note that you covered 180 m before turning around.

Analyzing the 90 second out-and-back trip, what is your average speed?

1. zero
2. 4 m/s
3. 4.5 m/s
4. 5 m/s
5. 9 m/s

Average speed is the total distance covered divided by the total time. That works out to 360 m / 90 s = 4 m/s

What is your average velocity?

1. zero
2. 1.5 m/s West
3. 3 m/s West
4. 4 m/s West
5. none of the above

Average velocity is the net displacement divided by the total time. Since the net displacement is zero, v_avg = 0.

• 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

Instantaneous acceleration: a_x = dv/dt

Average acceleration = a_avg = Δv/Δt

Δv_x = ∫ a_x dt