1.1 Scalars and Vectors in One Dimension

Scalars have only magnitude, like distance and speed. Vectors have both magnitude and direction, like position, displacement, velocity, and acceleration. In AP Physics 1, separating scalar and vector quantities helps you track direction correctly in one-dimensional motion.

Why This Matters for the AP Physics 1 Exam

This topic is the foundation for all of kinematics and most of AP Physics 1. Once you can tell a scalar from a vector and use signs to track direction, you can correctly add velocities, find displacement, and set up motion problems without sign errors.

The exam rewards clear use of representations. Vectors get drawn as arrows, written with signs, and connected to equations, and the AP exam asks you to move between those forms. That skill shows up across the multiple-choice section and in the free-response question that focuses on translating between representations.

Key Takeaways

• A scalar has magnitude only (distance, speed); a vector has magnitude and direction (position, displacement, velocity, acceleration).
• In one dimension, a positive or negative sign fully describes a vector's direction along the chosen axis.
• Vectors can be drawn as arrows where the arrow length is proportional to the magnitude.
• Full vectors are written with an arrow above the symbol, like $\vec{v}=\vec{v}_0+\vec{a}t$; one-dimensional components drop the arrow and use signs, like $v_x=v_{x0}+a_xt$.
• To add vectors in one dimension, add their signed values; same direction adds, opposite directions subtract.
• Distance is a scalar that counts all path length; displacement is a vector equal to final position minus initial position.

Scalar and Vector Quantities

Scalars vs Vectors

• Scalars are described by magnitude only. Distance and speed are scalars.
• Vectors are described by both magnitude and direction. Position, displacement, velocity, and acceleration are vectors.
• A vector can be drawn as an arrow. The direction of the arrow shows the direction of the quantity, and the length is proportional to the magnitude.

Example: "She is five feet tall" gives only a magnitude, so it is a scalar description.

Example: "The gas station is five miles west" gives both a magnitude and a direction, so it describes a vector.

Position is a vector because it describes an object's location relative to an origin in a chosen coordinate system. In one dimension, position can be positive or negative depending on which side of the origin the object is on.

When you draw vectors as arrows, the arrow length should match the magnitude. A 5 m arrow is shorter than a 50 m arrow to show the difference in size.

Key points to remember about scalars and vectors:

• Scalars can be combined with simple arithmetic. Vectors require vector addition or subtraction.
• Vectors carry a direction; scalars do not.
• A scalar is a single number; a vector needs a magnitude and a direction.
• Vectors are usually marked with an arrow above the symbol, such as $\vec{v}$. In one dimension, the component can be written without an arrow, such as $v_x$, because the sign shows the direction.

Vector Notation

• Position, displacement, velocity, and acceleration are vector quantities.
• Full vector notation uses an arrow above the symbol, such as $\vec{v}$ for velocity, and a vector equation looks like $\vec{v}=\vec{v}_0+\vec{a}t$.
• In one dimension, the arrow is often dropped for components along an axis, so the same relationship can be written as $v_x=v_{x0}+a_xt$.
• The sign of the component completely describes its direction along the axis: positive one way, negative the other.

Examples of Scalars and Vectors

Scalar examples (magnitude only):

• Distance traveled during a trip (300 miles)
• Speed of a car on the highway (65 mph)
• Mass of an object (5 kg)
• Temperature outside (75°F)

Vector examples (magnitude and direction):

• Position of a mailbox 200 m east of an origin (+200 m)
• Displacement from start to end (50 km east)
• Velocity of a train moving in a straight line (500 mph east)
• Acceleration due to gravity acting downward ($-9.8 \text{ m/s}^2$)
• Force pushing a box (20 N westward)

Distance vs Displacement

Distance and displacement are easy to mix up, so keep them separate.

Displacement is a vector equal to the change in position.

• It is calculated as final position minus initial position.
• It can be smaller than the actual distance traveled if the path is not straight.
• It can be zero if an object returns to its starting point.

Distance is a scalar equal to the total path length traveled.

• It counts every part of the path, no matter the direction.
• It is always positive and is at least as large as the magnitude of displacement.
• For a round trip, distance is the sum of all segments traveled.

Vector Sum in One Dimension

Opposite Directions and Signs

• In a one-dimensional coordinate system, opposite directions are shown by opposite signs.
  • Rightward or upward is usually positive.
  • Leftward or downward is usually negative.
• Find a vector sum by adding the signed values of the individual vectors.
  • Same direction: add the magnitudes (3 m/s + 5 m/s = 8 m/s).
  • Opposite directions: subtract to get the difference (5 m/s - 3 m/s = 2 m/s, in the direction of the larger one).

How to Use This on the AP Physics 1 Exam

Problem Solving

1. Pick a positive direction before you do any math, and label it (for example, east is +).
2. Write each vector with the correct sign based on that choice.
3. Add the signed values to get the resultant. The sign of your answer tells you the final direction.
4. State the answer with both magnitude and direction.

Common Trap

Be careful not to treat speed and velocity as the same thing. Speed is a scalar with no direction; velocity is a vector that needs a sign or direction. Mixing them up causes sign errors when you add or compare motion.

Practice Problem 1: Vector Addition in One Dimension

Step 1: Set up the coordinate system.

• Let east be positive (+) and west be negative (-).

Step 2: Calculate the displacement.

• First displacement: +35 km (east)
• Second displacement: -20 km (west)
• Total displacement = +35 km + (-20 km) = +15 km

Step 3: Interpret the result.

• The final displacement is 15 km east of the starting point.

Step 4: Calculate average velocity.

• Average velocity = total displacement ÷ total time
• Average velocity = 15 km ÷ 1 hour = 15 km/h east

The final displacement is 15 km east, and the average velocity is 15 km/h east.

Practice Problem 2: Vector Sum of Displacements

Step 1: Set up the coordinate system.

• Let east be positive (+) and west be negative (-).

Step 2: Calculate the vector sum.

• First displacement: +12 m (east)
• Second displacement: -7 m (west)
• Total displacement = +12 m + (-7 m) = +5 m

Step 3: Interpret the result.

• The resultant displacement is 5 m east.

This shows how opposite signs represent opposite directions and how adding signed values gives the vector sum.

Practice Problem 3: Scalar vs Vector Quantities

Step 1: Calculate the total distance (scalar).

• First leg: 3 km
• Second leg: 4 km
• Third leg: 3 km
• Total distance = 3 km + 4 km + 3 km = 10 km

Step 2: Calculate the resultant displacement (vector).

• Let east be positive (+) and west be negative (-).
• Displacements: +3 km, -4 km, +3 km
• Total displacement = +3 + (-4) + (+3) = +2 km

Step 3: Interpret the result.

• The runner's displacement is 2 km east.

Even though the runner covered 10 km of distance, the displacement is only 2 km east. That gap is the core difference between a scalar and a vector.

Common Misconceptions

• Distance and displacement are not the same. Distance adds up all path length; displacement only cares about start and end positions, including direction.
• Speed and velocity are different. Speed is a scalar; velocity is a vector with direction.
• A negative sign on a vector does not always mean "slowing down." In one dimension, the sign just means direction along the axis, like west or downward.
• Vector notation is not about uppercase versus lowercase letters. Full vectors get an arrow above the symbol, such as $\vec{v}$, while a one-dimensional component like $v_x$ uses its sign to show direction.
• A larger distance does not guarantee a larger displacement. If the path curves or doubles back, displacement can be much smaller than distance, or even zero.

Related AP Physics 1 Guides

• 1.2 Displacement, Velocity, and Acceleration
• 1.3 Representing Motion
• 1.4 Reference Frames and Relative Motion
• 1.5 Vectors and Motion in Two Dimensions