1.1 Scalars and Vectors in One Dimension

Scalars and vectors are fundamental concepts in physics, describing quantities with different characteristics. Scalars only have magnitude, while vectors have both magnitude and direction, crucial for understanding motion and forces.

In one dimension, vectors can be represented using positive and negative signs to indicate direction. This simplifies calculations and allows for easy addition or subtraction of vectors along a single axis, essential for solving physics problems.

Scalar and vector quantities

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Scalars vs vectors

• Scalars quantities described solely by magnitude 📏 (distance, speed)
• Vectors quantities described by both magnitude and direction 🧭 (position, displacement, velocity, acceleration)
• Visual representation of vectors arrows with appropriate direction and lengths proportional to magnitude

Example: She is five feet tall

• Distance and Speed are scalar quantities

Key Vocabulary: Vector - quantities that are described by a size (magnitude) and a direction (ex. East, Up, Right, etc.)

Example: The gas station is five miles west from the car

• Position, Displacement, Velocity, and Acceleration are vector quantities

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

Vectors can also be represented by arrows, and the length of the arrow should represent the magnitude of the described quantity. From the image below you can see the 5m arrow is smaller in length than the 50m arrow to reflect the difference in magnitude of the two quantities.

Here are some key points to remember about the difference between scalar and vector quantities in AP Physics 1:

• Scalar quantities can be added or subtracted using simple arithmetic, but vector quantities require the use of vector addition or subtraction.
• Vector quantities have a direction associated with them, while scalar quantities do not.
• Scalar quantities are described by a single number, while vector quantities are described by both a magnitude and a direction.
• Examples of scalar quantities include mass, volume, density, and temperature. Examples of vector quantities include position, displacement, velocity, acceleration, and force.
• Scalar quantities can be represented by a single point on a number line, while vector quantities are represented by an arrow with a magnitude and a direction.
• Scalars and vectors are not distinguished by uppercase or lowercase letters. Vectors are commonly indicated with an arrow above the symbol, such as $\vec{v}$, while in one dimension the component may be written without an arrow and the sign indicates direction, such as $v_x$.

Vector representation

• Position, displacement, velocity, acceleration vector quantities
  • Notation arrow above the symbol for quantity ($\vec{v}$ for velocity)
  • Vectors may be written with arrows above the symbols, such as $\vec{v}=\vec{v}_0+\vec{a}t$. In one dimension, vector arrows are often omitted for components along an axis because the sign gives the direction, so the same relationship can be written as $v_x=v_{x0}+a_xt$.
  • Components along an axis do not require vector notation in one dimension
    • Sign of component completely describes direction along the axis (positive or negative)

Examples of scalars and vectors

Understanding the distinction between scalars and vectors becomes clearer with practical examples:

Scalar examples show only magnitude:

• 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 show both magnitude and direction:

• Position of a mailbox located 200 m east of an origin (+200 m)
• Displacement from starting point to ending point (50 km east)
• Velocity of a train moving at a constant speed in a straight line (500 mph east)
• Acceleration due to gravity acting downward on an object (-9.8 m/s²)
• Force applied to push a box (20 N westward)

Displacement and distance are two important concepts in physics that are often confused with one another.

Displacement is the straight-line distance between the starting and ending positions, considering direction.

• It represents the change in position of an object
• Displacement is calculated as final position minus initial position
• It can be smaller than the actual distance traveled if the path isn't straight
• It can even be zero if an object returns to its starting point

Distance is the total length of the path traveled, regardless of direction changes.

• It represents the total ground covered during movement
• Distance is always positive and is at least as large as displacement
• It accounts for every twist and turn in the path
• 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 denoted by opposite signs ➕➖
  • Rightward or upward typically positive
  • Leftward or downward typically negative
• Determine vector sum by adding the signed magnitudes of the individual vectors
  • Two vectors pointing in the same direction sum of their magnitudes (3 m/s + 5 m/s = 8 m/s)
  • Two vectors pointing in opposite directions difference of their magnitudes (5 m/s - 3 m/s = 2 m/s)

Practice Problem 1: Vector Addition in One Dimension

To solve this problem, we need to find the vector sum of the two displacements.

Step 1: Assign directions to our coordinate system.

• Let east be the positive direction (+)
• Let west be the negative direction (-)

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 car's final displacement is 15 km east of the starting point, and its average velocity is 15 km/h east.

Practice Problem 2: Vector Sum of Displacements

Step 1: Assign directions to our coordinate system.

• Let east be the positive direction (+)
• Let west be the negative direction (-)

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 problem shows how opposite signs represent opposite directions in one dimension, and how adding signed values gives the vector sum.

Practice Problem 3: Scalar vs. Vector Quantities

Step 1: Calculate the total distance traveled (scalar quantity).

• 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 quantity).

• Let east be the positive direction (+) and west be the negative direction (-)
• First displacement: +3 km (east)
• Second displacement: -4 km (west)
• Third displacement: +3 km (east)
• Total displacement = +3 + (-4) + (+3) = +2 km

Step 3: Interpret the result.

• The runner's displacement is 2 km east.

This problem illustrates the important difference between distance (a scalar) and displacement (a vector). Even though the runner traveled a total distance of 10 km, their displacement from the starting point is only 2 km east.