1.1 - Scalars and Vectors in One Dimension

Scalar Quantities

Scalars are physical quantities that only have magnitude, meaning just a numerical size or amount, with no direction attached.

For example, if someone walks 45 meters, that tells you how far they went but not which way. That value could apply to movement in literally any direction.

Common scalars in kinematics: distance, speed, time

Other scalars in physics: energy, power, mass, temperature

Vector Quantities

Vectors need both magnitude and direction to fully describe them. That extra directional piece is what separates them from scalars.

For example, walking 45 meters north gives you the complete picture: how far someone went and exactly which way.

Common vectors in kinematics: position, displacement, velocity, acceleration

Other vectors in physics: force, momentum, electric field, magnetic field

Visualizing Vectors

Vectors are drawn as arrows, where:

the length of the arrow shows the magnitude

the direction of the arrow shows the direction

The tail is where the vector starts, and the head is the arrowhead at the end.

1.1 - Scalars and Vectors in One Dimension

Drawing vectors to scale: pick a scale and use it for every arrow in the same diagram. If 1 cm stands for 5 m/s, a velocity of 20 m/s is drawn 4 cm long, and a velocity of 10 m/s in the opposite direction is drawn 2 cm long, pointing the other way. A vector with twice the magnitude always gets an arrow twice as long.

Vector Notation

You write a vector with a little arrow above the symbol, so

\vec{v}

means velocity as a vector.

When you’re just working in one dimension, subscripts tell you which direction you’re dealing with:

$v_x$ for horizontal velocity

$v_y$ for vertical velocity

$a_x$ for horizontal acceleration

The subscript is basically just a reminder of which axis you’re looking at.

Vector form vs. component form: written as a full vector equation, the velocity relationship is

\vec{v} = \vec{v}_0 + \vec{a}t

. The AP equation sheet writes the same relationship along one axis:

v_x = v_{x0} + a_x t

Components don’t need arrows. A component like

v_x

a_x

is just a number with a sign and a unit. In one dimension the sign already tells you the direction, so no arrow is needed.

Example: a cart rolls in the

+x

direction at

v_{x0} = +2.0 \text{ m/s}

and has a constant acceleration of

a_x = -0.50 \text{ m/s}^2

. After 6.0 s:

v_x = v_{x0} + a_x t = (+2.0) + (-0.50)(6.0) = -1.0 \text{ m/s}

The negative sign by itself tells you the cart has turned around and is now moving at 1.0 m/s in the

-x

direction.

Sign Convention in One Dimension

In one dimension, direction gets shown through positive and negative signs:

positive usually means right or up

negative usually means left or down

For example:

$v = +30 \text{ m/s}$ means moving 30 m/s in the positive direction

$v = -30 \text{ m/s}$ means moving 30 m/s in the negative direction

Adding and Subtracting Vectors

Combining vectors to get one resultant vector is called a vector sum, and this covers both addition and subtraction.

Addition:

\vec{A} + \vec{B} = \vec{C}

Subtraction:

\vec{A} - \vec{B} = \vec{C}

Subtracting a vector is really the same as adding its opposite:

\vec{A} - \vec{B} = \vec{A} + (-\vec{B})

In one dimension, signs do the work. Choose a positive direction, give every vector a

+

-

sign to match its direction, then add the signed numbers. Opposite directions always get opposite signs.

Currents in real life:

When a current flows the same way you’re moving, the velocities just add together (downstream). When the current works against you, it subtracts from your velocity (upstream).

In competitive sprinting, there’s actually a rule about this: if the wind pushing a runner along is stronger than the allowed limit (usually 2.0 m/s), the record doesn’t count because the runner got too much help from the wind. Headwinds, on the other hand, always count toward the record since they make the performance more impressive, not less.

Worked Example: Vector Sum in One Dimension

A robot vacuum moves 3.5 m east, then 6.0 m west, then 1.2 m east. Find its total displacement and the total distance it traveled.

Setup: east is positive, so the three displacements are

+3.5

-6.0

m, and

+1.2

Displacement (vector sum):

\Delta x_{\text{total}} = (+3.5) + (-6.0) + (+1.2) = -1.3 \text{ m}

The negative sign means the robot ends up 1.3 m west of where it started.

Distance (scalar sum): distance ignores direction, so just add the sizes:

d = 3.5 + 6.0 + 1.2 = 10.7 \text{ m}

Subtracting vectors works the same way. If a ball hits a wall at

+8.0

m/s and bounces straight back at

-6.0

m/s, its change in velocity is

\Delta v = v - v_0 = (-6.0) - (+8.0) = -14.0 \text{ m/s}

That’s bigger than either speed, because reversing direction means the ball first loses all 8.0 m/s and then gains 6.0 m/s the other way.

Rate, Distance, and Time

The basic relationship between speed, distance, and time is:

r = \frac{d}{t}

where

r

is rate (speed),

d

is distance, and

t

is time.

You can rearrange this to solve for time. Starting with

r = \frac{d}{t}

, multiply both sides by

t

to get

r \cdot t = d

, then divide both sides by

r

t = \frac{d}{r}

That gives you time whenever you know distance and rate.

Worked Example: River Kayaking

Two kayakers start at the same time from opposite ends of a 24-mile river route. Both can paddle at 6 mph in still water. Kayaker A goes downstream (with the current), Kayaker B goes upstream (against it), and the current itself is 2 mph. How long does each one take, and how much sooner does one finish?

Using

t = \frac{d}{r}

Kayaker A (downstream): the current helps, so their effective speed is

6 + 2 = 8

mph.

t_A = \frac{24}{8} = 3.0 \text{ hours}

Kayaker B (upstream): the current fights them, so their effective speed is

6 - 2 = 4

mph.

t_B = \frac{24}{4} = 6.0 \text{ hours}

Difference:

\Delta t = t_B - t_A = 6.0 - 3.0 = 3.0 \text{ hours} = 180 \text{ minutes}

So Kayaker A finishes a full 3 hours (180 minutes) before Kayaker B.

Scalars vs. Vectors at a Glance

Scalar	Vector
Magnitude only	Magnitude and direction
Simple arithmetic	Follows vector addition rules
Example: 15 meters, 8 seconds	Example: 15 m east, 8 m/s upward
No directional symbol	Arrow notation $\vec{v}$ or subscripts $v_x$

Key Takeaways

Scalars have magnitude only: distance, speed, time, energy

Vectors have magnitude and direction: displacement, velocity, acceleration, force

Vectors are drawn as arrows, arrow length shows magnitude, arrow direction shows direction

Vector notation uses arrows over letters ( $\vec{v}$ ) or subscripts ( $v_x$ , $v_y$ )

In one dimension, opposite directions get opposite signs

Vector subtraction just means adding the inverse: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Time formula: $r = \frac{d}{t} \rightarrow t = \frac{d}{r}$

Helpful currents/winds add to velocity, opposing ones subtract from it

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