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Scalars & Vectors
in One Dimension

Master the difference between magnitude alone and magnitude with direction: the foundation of every motion problem in physics.

AP Physics 1

What Is a Scalar?

A scalar is a physical quantity described by a single number: its magnitude. Scalars carry no information about direction. They can be positive, zero, or (in some cases) negative, but the sign does not indicate a spatial direction.

Mass
\( m = 5 \text{ kg} \): always positive, no direction needed.
Speed
\( |\vec{v}| = 12 \text{ m/s} \): how fast, but not which way.
Time
\( t = 3.0 \text{ s} \): duration only, no direction.

What Is a Vector?

A vector is a physical quantity that has both magnitude and direction. In one dimension, direction is encoded by the sign: positive for one way, negative for the other.

\( +8 \text{ m/s} \) \( -5 \text{ m/s} \) 0 +x −x

Why Does the Distinction Matter?

Treating a vector as a scalar (or vice versa) leads to wrong answers. Every kinematics equation in AP Physics 1 uses vector quantities for position, velocity, and acceleration.

1
Scalars add like ordinary numbers. \( 5 \text{ kg} + 3 \text{ kg} = 8 \text{ kg} \).
2
Vectors add with signs. \( +4 \text{ m/s} + (-6 \text{ m/s}) = -2 \text{ m/s} \).
3
Ignoring the sign can flip the direction of your answer entirely.

The Core Rule: Sign = Direction

+ direction − direction Origin \( v = +6 \text{ m/s} \) \( v = -4 \text{ m/s} \)

Position, Displacement, & Distance

Position \( x \)
Vector. Where an object sits on the axis relative to the origin. Signed.
Displacement \( \Delta x \)
Vector. Change in position: \( \Delta x = x_f - x_i \). Signed.
Distance \( d \)
Scalar. The total path length traveled. Always \( \geq 0 \).

Velocity vs Speed

Velocity
Vector. \( v = \dfrac{\Delta x}{\Delta t} \). Has a sign indicating direction.
Speed
Scalar. \( \text{speed} = |v| \). Magnitude of velocity. Always \( \geq 0 \).
-3 -2 -1 0 +1 +2 \( v = -3 \text{ m/s} \) \( \text{speed} = 3 \text{ m/s} \)

Acceleration in 1-D

Acceleration is a vector: \( \vec{a} = \dfrac{\Delta v}{\Delta t} \). Its sign tells you the direction of the velocity change, not necessarily the direction of motion.

Speeding Up
\( v \) and \( a \) have the same sign. Object gets faster.
Slowing Down
\( v \) and \( a \) have opposite signs. Object decelerates.

Scalar vs Vector: Side by Side

Scalar Vector (1-D)
Has magnitude? ✓ Yes ✓ Yes
Has direction? ✕ No ✓ Yes (sign)
Can be negative? Depends (e.g., temperature) ✓ Always (sign = direction)
Examples mass, speed, distance, time, energy position, displacement, velocity, acceleration, force
Addition Ordinary arithmetic Sign-aware arithmetic

How to Apply the Sign Convention

1
Choose an axis. Draw a straight line and label one direction as positive (e.g., right = +x).
2
Assign signs. Any quantity pointing in the + direction gets a positive value; opposite gets negative.
3
Use signed arithmetic. Substitute signed values into equations: \( \Delta x = x_f - x_i \), \( a = \frac{\Delta v}{\Delta t} \).
4
Interpret the result. Positive result = + direction. Negative result = − direction.
5
Extract scalars if needed. Take the absolute value for speed or distance: \( \text{speed} = |v| \).

Example 1: Displacement & Distance

A jogger runs 200 m east, then turns and runs 80 m west. Let east = +x.

+x (east) Start +200 m Turn −80 m End

Example 2: Velocity from Position

A ball starts at \( x_i = +5 \text{ m} \) and reaches \( x_f = -3 \text{ m} \) in \( \Delta t = 2 \text{ s} \). Find the velocity and speed.

1
Displacement: \( \Delta x = x_f - x_i = (-3) - (+5) = -8 \text{ m} \)
2
Velocity: \( v = \dfrac{\Delta x}{\Delta t} = \dfrac{-8}{2} = \) \( -4 \text{ m/s} \) (negative direction)
3
Speed: \( |v| = |-4| = \) \( 4 \text{ m/s} \)

Example 3: Acceleration Sign

A car moves right (\( v_i = +20 \text{ m/s} \)) and brakes to \( v_f = +8 \text{ m/s} \) in \( 4 \text{ s} \). Find the acceleration. Is the car speeding up or slowing down?

1
\( a = \dfrac{v_f - v_i}{\Delta t} = \dfrac{8 - 20}{4} = \dfrac{-12}{4} = \) \( -3 \text{ m/s}^2 \)
2
\( v \) is positive, \( a \) is negative: opposite signs means slowing down.

Adding Vectors in 1-D

To add vectors in one dimension, simply add their signed values. Place the tail of the second vector at the tip of the first.

\( \vec{A} = +6 \) \( \vec{B} = -2 \) \( \vec{R} = +4 \)

Common Mistakes to Avoid

✕ Dropping the Sign
Writing \( v = 5 \text{ m/s} \) when the object moves left. You need \( v = -5 \text{ m/s} \).
✕ Distance = Displacement
Distance is the total path length (always positive). Displacement can be negative.
✕ Negative \( a \) = Slowing Down
Negative acceleration means the force points in the − direction. It could be speeding up if \( v \) is also negative.

Summary of Key Quantities

Quantity Type Symbol SI Unit
Position Vector \( x \) m
Displacement Vector \( \Delta x \) m
Distance Scalar \( d \) m
Velocity Vector \( v \) m/s
Speed Scalar \( |v| \) m/s
Acceleration Vector \( a \) \( \text{m/s}^2 \)
Mass Scalar \( m \) kg
Time Scalar \( t \) s

Question 1 of 5

Which of the following is a vector quantity?

Question 2 of 5

An object moves from \( x_i = +4 \text{ m} \) to \( x_f = -6 \text{ m} \). What is the displacement?

Question 3 of 5

A car has velocity \( v = -12 \text{ m/s} \) and acceleration \( a = -3 \text{ m/s}^2 \). Is the car speeding up or slowing down?

Question 4 of 5

A runner goes 50 m north, then 30 m south (let north = +). What are the runner's distance and displacement?

Question 5 of 5

Two forces act on a box along the x-axis: \( F_1 = +15 \text{ N} \) and \( F_2 = -9 \text{ N} \). What is the net force?