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Scalars and Vectors in One Dimension

Lesson ~10 min read 8 MCQs

In simple terms: In simple terms, scalars and vectors are about how we describe motion. Scalars just tell you "how much" (like distance), while vectors tell you "how much" and "which way" (like displacement).

Why this matters

Imagine you're in downtown Boston, and you ask someone for directions to a famous pizza place. They say, "Oh, it's easy! Just walk 2 miles." You start walking, but after 30 minutes, you're completely lost. Why? Because you knew how far to walk, but not in which direction. That "2 miles" is a scalar—it has a size (a magnitude), but no direction.

What you really needed was a vector: "Walk 2 miles northeast on Hanover Street." That piece of information has both a magnitude (2 miles) and a direction (northeast). In physics, this distinction is everything. It's the difference between knowing the total steps on your fitness tracker and knowing how to get from your starting point to your destination. Today, we'll master this fundamental concept for motion in one dimension.

Concept overview

flowchart TD
    A[Physical Quantity] --> B{Does it have direction?}
    B -- Yes --> C[Vector]
    B -- No --> D[Scalar]
    C --> E[Examples:<br/>Displacement<br/>Velocity<br/>Acceleration<br/>Force]
    D --> F[Examples:<br/>Distance<br/>Speed<br/>Mass<br/>Time]

Core explanation

Welcome to kinematics! This is the branch of physics that describes motion. Before we can analyze a car accelerating or a ball being thrown, we need to agree on a language to describe where things are and how they're moving. That language starts with scalars and vectors.

What's a Scalar?

A scalar is the simplest type of measurement you can imagine. It's just a number with its units. It tells you "how much" of something there is.

Think about it:

  • The temperature outside is 75°F.
  • A bag of apples costs $8.
  • The game is 60 minutes long.

These values have a magnitude (75, 8, 60), but no direction. It wouldn't make sense to say "the temperature is 75°F to the left." That's a scalar.

In physics, two key scalars we'll use right away are distance and speed.

Distance is the total amount of ground an object has covered. It's like the reading on a car's odometer or the step count on your phone. It just adds up, no matter which direction you go.

What's a Vector?

A vector is a more complete description. It has a magnitude (how much) AND a direction (which way).

If you tell someone you're applying a 50-pound force, they'll ask, "in what direction?" Are you pushing a box forward, lifting it up, or pulling it sideways? The direction matters! Force is a vector.

In kinematics, our most important vectors are:

  • Position
    (x): Where an object is located relative to a starting point (the origin).
  • Displacement
    (Δx): The change in position. It's a straight line from the start point to the end point.
  • Velocity
    (v): How fast and in which direction an object is moving.
  • Acceleration
    (a): The rate of change of velocity.

Distance vs. Displacement: A Tale of Two Paths

Imagine a number line drawn on the floor. Let's say you start at the origin (position x = 0 m).

  1. You walk 5 meters in the positive direction (to the right) to position x = +5 m.
  2. Then, you turn around and walk 3 meters back in the negative direction (to the left), ending up at position x = +2 m.

Let's break this down:

  • Total Distance (a scalar)
    You walked 5 meters, then another 3 meters. Your feet covered a total of 5 m + 3 m = 8 m. Distance is always positive and just adds up.
  • Net Displacement (a vector)
    Displacement only cares about your starting point and your ending point. You started at x = 0 m and ended at x = +2 m. Your displacement is your final position minus your initial position: Δx = x_final - x_initial = 2 m - 0 m = +2 m

The displacement is "+2 meters." The magnitude is 2 meters, and the direction is given by the positive sign, meaning "in the positive direction from where you started."

Think of it this way: Distance is the scenic route your car actually drove. Displacement is the "as the crow flies" straight-line distance from your driveway to your destination.

Signs as Directions in One Dimension

In AP Physics 1, we often deal with motion along a straight line (one dimension). This could be horizontal (left/right) or vertical (up/down). We have a fantastic shortcut for direction in 1D: positive and negative signs.

We get to define our coordinate system. Usually, we say:

  • Right
    is the positive (+) direction.
  • Left
    is the negative (-) direction.
  • Up
    is the positive (+) direction.
  • Down
    is the negative (-) direction.

So, a displacement of -3 m doesn't mean you moved a negative amount. It means you moved 3 meters in the negative direction (e.g., to the left).

Vector Addition in 1D

Let's go back to our walking example. We can also find the net displacement by adding the individual displacement vectors.

  • First movement: Δx₁ = +5 m (5 m to the right)
  • Second movement: Δx₂ = -3 m (3 m to the left)

The total (or net) displacement is the vector sum: Δx_total = Δx₁ + Δx₂ = (+5 m) + (-3 m) = +2 m

See how that works? Adding vectors in one dimension is just adding signed numbers. This is a powerful and simple tool.

Speed vs. Velocity

This same scalar/vector distinction applies to speed and velocity.

  • Average Speed (scalar)
    Total distance / time elapsed.
  • Average Velocity (vector)
    Displacement / time elapsed.

Let's say your walk (5 m right, 3 m left) took a total of 4 seconds.

  • Average Speed
    = (8 m) / (4 s) = 2 m/s.
  • Average Velocity
    = (+2 m) / (4 s) = +0.5 m/s.

Your average velocity is +0.5 m/s, meaning that on average, you made progress in the positive direction at a rate of half a meter per second. Your speed was much higher because it accounts for all the back-and-forth motion.

When we write vector quantities like velocity or acceleration, the formal notation is an arrow over the symbol, like v⃗. But in 1D problems, you'll almost always see it written as a component, like vₓ, where the sign tells you the direction. Don't let the different notations throw you; they mean the same thing in this context.

Worked examples

Let's put these ideas into practice. The key is to read the problem carefully and distinguish what it's asking for: a scalar (like distance) or a vector (like displacement).

Example 1

The Track Runner

Problem: Aaliyah runs on a straight track. She starts at the 0-meter line, runs 100 meters east, and then immediately turns around and jogs 40 meters west. What is her total distance traveled, and what is her net displacement?

Solution:

  1. 1
    Define a coordinate system
    This is always step one! Let's say "east" is the positive direction and "west" is the negative direction. Her starting point is x_initial = 0 m.
  2. 2
    Calculate the total distance (scalar)
    Distance is the sum of the lengths of all paths taken, regardless of direction.
    • Path 1: 100 m
    • Path 2: 40 m
    • Total Distance = 100 m + 40 m = 140 m
    • Notice that distance is just a positive number.
  3. 3
    Calculate the net displacement (vector)
    Displacement is the vector sum of the individual movements.
    • Displacement 1 (Δx₁): 100 m east = +100 m
    • Displacement 2 (Δx₂): 40 m west = -40 m
    • Net Displacement (Δx_total) = Δx₁ + Δx₂
    • Δx_total = (+100 m) + (-40 m) = +60 m

Why this matters: Aaliyah's fitness tracker would say she traveled 140 meters. But if you were measuring her progress from the start line, she only ended up 60 meters away from it, in the positive (east) direction. Both numbers are correct, but they describe different things.

Example 2

The Elevator Ride

Problem: An elevator in a tall building starts at the ground floor (0 m). It travels up to the 20th floor, at a height of 80 m. It then travels down to the 15th floor, at a height of 60 m. The entire trip takes 50 seconds. Find the elevator's (a) total distance traveled, (b) net displacement, (c) average speed, and (d) average velocity.

Solution:

  1. 1
    Define the coordinate system
    Let's set "up" as the positive direction and the ground floor as the origin (y = 0 m).
  2. 2
    (a) Total Distance (scalar)
    • Upward travel: It went from 0 m to 80 m, so it traveled 80 m.
    • Downward travel: It went from 80 m to 60 m, so it traveled 80 m - 60 m = 20 m.
    • Total Distance = 80 m + 20 m = 100 m
  3. 3
    (b) Net Displacement (vector)
    • We only need the initial and final positions.
    • Initial Position (y_initial): 0 m (ground floor)
    • Final Position (y_final): 60 m (15th floor)
    • Net Displacement (Δy) = y_final - y_initial = 60 m - 0 m = +60 m
    • The "+" sign means the final position is 60 meters above the starting point.
  4. 4
    (c) Average Speed (scalar)
    • Average Speed = Total Distance / Time
    • Average Speed = 100 m / 50 s = 2 m/s
  5. 5
    (d) Average Velocity (vector)
    • Average Velocity = Net Displacement / Time
    • Average Velocity = +60 m / 50 s = +1.2 m/s

Try it yourself

Time to try one on your own. Don't worry about getting it perfect, just focus on applying the process.

Problem: A small quadcopter drone takes off from a patio table. It flies straight up 12 meters to get above a tree. A gust of wind then pushes it straight down 15 meters, where it hovers.

  1. What is the total distance the drone traveled?
  2. What is the drone's final displacement relative to the tabletop?

Hints:

  • Start by defining your coordinate system. Which way is positive? Up or down?
  • For displacement, remember that it's final position - initial position. Where did it start? Where did it end up?
  • Pay close attention to your signs!
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