A1.12 Use numeric, symbolic, graphical, and vector representations

Ontario Grade 12 Physics

Scientific Investigation Skills and Career Exploration

Moving between numeric, symbolic, graphical, and vector representations

In physics, the same situation can be described with numbers, symbols, a graph, or arrows. Each representation makes some features easier to see. Numbers can show measured or calculated values. Symbols show a relationship that can apply to many cases. A graph shows how quantities relate or change. Vectors show both a quantity’s size and its direction. A strong solution can move between these forms without changing what the situation means. This lesson practises those translations using familiar motion and force examples.

What you will learn

Identify numeric, symbolic, graphical, and vector representations of a physical situation.
Translate information between representations while keeping the system, frame, direction, and units consistent.
Use a graph or vector diagram to communicate information that numbers and equations alone may hide.
Check whether different representations describe the same physical situation.

1. Begin with the system, frame, and direction

A physical system is the object or group of objects being described. A reference frame is the viewpoint used to describe position and motion. For everyday problems, the frame is often the ground or classroom. State it so that words such as “moving” have a clear meaning.

Choose a positive direction before using signed numbers or vector components. For example, take right as positive and left as negative. A negative value then means the quantity points left; it does not mean that the quantity has an impossible size.

A scalar has magnitude, or size, but no direction. Time and distance are scalars. A vector has both magnitude and direction. Displacement, velocity, and force are vectors. In one dimension, a vector can be represented by a signed number once a positive direction is chosen.

Name the object, reference frame, and positive direction.
Scalars have size only; vectors have size and direction.
A sign communicates direction only after the positive direction is defined.

2. Four representations and what they show

A numeric representation uses values with units, such as a displacement of

+12\ \mathrm{m}

. The unit identifies the kind of quantity and scale. The plus sign identifies direction under the stated convention.

A symbolic representation uses letters and operations to express a general relationship. For constant velocity in one dimension, displacement equals velocity multiplied by elapsed time. The symbols can be used with different values, provided the quantities and units are appropriate.

A graphical representation places quantities on labelled axes. For a position–time graph, time belongs on the horizontal axis and position on the vertical axis. A straight segment that rises as time increases represents motion in the positive direction. Its steepness shows how quickly position changes. The slope of a straight position–time segment is velocity.

A vector representation uses an arrow. The arrow’s direction shows the vector’s direction, and its length represents magnitude according to a chosen scale. For example, an arrow to the right labelled

12\ \mathrm{m}

represents a displacement of 12 m to the right. In one dimension, the same information can be written as

+12\ \mathrm{m}

when right is positive.

These forms are not separate answers. They should agree. A graph rising to the right, a positive velocity, and a right-pointing displacement arrow can all describe motion in the positive direction.

v=\frac{\Delta x}{\Delta t}

Label graph axes and include units.
Use a vector arrow when direction matters.
Translate meaning, not just notation: preserve the quantity, magnitude, direction, and units.

3. Translate carefully and check for agreement

The symbol

\Delta

means “change in.” For position, the change is final position minus initial position. A graph gives the same change by comparing the vertical values at two times. The velocity for a straight segment can then be found by dividing the position change by the time change.

A graph is especially useful when a situation includes more than one interval. A change in slope signals a change in velocity. A horizontal position–time segment shows that position is constant during that interval, so the object is at rest relative to the chosen frame.

Vectors can also be combined. In one dimension, choose a positive direction and add signed components. For example, a 9 m displacement right followed by a 4 m displacement left gives a net displacement of 5 m right. The result is a vector: it needs both magnitude and direction.

Before accepting a representation, check its labels and units. A position–time graph should not put position on the horizontal axis if it is meant to show position as a function of time. A vector answer should not give a direction that conflicts with its sign convention. A numeric result should also make sense compared with the values in the situation.

\Delta x=x_f-x_i

For a straight position–time segment, velocity is the change in position divided by the change in time.
Add one-dimensional vectors using signs set by the chosen positive direction.
Use units, signs, labels, and a reasonableness check to compare representations.

4. Build a consistent set of representations

A useful workflow is to define the system and frame, set a positive direction, identify the known quantities and the unknown, and choose a representation that suits the question. Then translate to another form as needed.

For a calculation, write the governing relationship before substituting values. Keep units in the substitution and report a sensible number of significant figures. For a graph, label axes and mark enough points to show the stated information. For a vector diagram, use arrows and labels that match the sign convention.

A model is a simplified description of a physical situation. Here, the model may be a straight position–time segment or a one-dimensional vector sum. The representation is useful only if its assumptions match the stated situation. Do not infer extra details that the information does not provide.

Define the situation before choosing symbols or drawing.
Check that every representation tells the same physical story.
Do not claim more detail than the values, graph, or diagram supports.

Worked example

From numbers to an equation and a graph

A cart moves in a straight line along a level track. The system is the cart, and the reference frame is the track. Right is positive. At

t=2.0\ \mathrm{s}

, its position is

x=3.0\ \mathrm{m}

. At

t=6.0\ \mathrm{s}

, its position is

x=15.0\ \mathrm{m}

. Find its velocity for this interval and describe matching numeric, symbolic, graphical, and vector representations.

Identify the changes
The final position and time are known. Subtract the initial values from the final values. The position change is positive, so the cart’s displacement is to the right under the stated convention.
$\Delta x=(15.0-3.0)\ \mathrm{m}=+12.0\ \mathrm{m}, \Delta t=(6.0-2.0)\ \mathrm{s}=4.0\ \mathrm{s}$
Use the relationship
For this straight position–time segment, velocity is displacement divided by elapsed time. The metres cancel with seconds in the denominator to give metres per second.
$v=\frac{\Delta x}{\Delta t}=\frac{+12.0\ \mathrm{m}}{4.0\ \mathrm{s}}=+3.0\ \mathrm{m/s}$
Describe the graph and vector
Plot time in seconds horizontally and position in metres vertically. Mark $(2.0\ \mathrm{s},3.0\ \mathrm{m})$ and $(6.0\ \mathrm{s},15.0\ \mathrm{m})$ , then join them with a straight rising segment. The vector form is an arrow to the right labelled $12.0\ \mathrm{m}$ for displacement, or a rightward velocity arrow labelled $3.0\ \mathrm{m/s}$ .

Answer: The cart’s velocity is

+3.0\ \mathrm{m/s}

, meaning

3.0\ \mathrm{m/s}

to the right. The numeric, symbolic, graphical, and vector descriptions agree.

Check: The units reduce to metres per second. The positive sign matches the rising graph and rightward arrow. A position increase of 12.0 m in 4.0 s makes 3.0 m/s reasonable.

Worked example

Read a position–time graph as a numeric relationship

A toy car moves along a straight floor. The system is the car, the reference frame is the floor, and right is positive. A straight position–time segment runs from

(0.0\ \mathrm{s}, 8.0\ \mathrm{m})

to

(5.0\ \mathrm{s}, -2.0\ \mathrm{m})

. Find the velocity and state its direction.

Read the endpoints
The graph gives initial and final positions as well as their times. The negative final position means the car is left of the chosen origin, not that its position is a negative distance.
$\Delta x=(-2.0-8.0)\ \mathrm{m}=-10.0\ \mathrm{m}, \Delta t=(5.0-0.0)\ \mathrm{s}=5.0\ \mathrm{s}$
Calculate the velocity
The segment is straight, so its velocity is constant over the interval. Divide the signed position change by the elapsed time.
$v=\frac{-10.0\ \mathrm{m}}{5.0\ \mathrm{s}}=-2.0\ \mathrm{m/s}$
Translate the sign
The negative result means the velocity points opposite to the positive direction. The graph descends as time increases, and a matching vector arrow points left.

Answer: The velocity is

-2.0\ \mathrm{m/s}

, or

2.0\ \mathrm{m/s}

to the left.

Check: The slope has units of metres per second and is negative because position decreases. The vector direction and the graph’s downward trend agree.

Worked example

Combine displacement vectors

A student walks 7.5 m east and then 2.0 m west along a straight hallway. Take the student as the system, use the hallway as the reference frame, and define east as positive. Find the net displacement in numeric, symbolic, and vector forms.

Assign signed components
East is positive, so the eastward displacement is positive. West is opposite to east, so the westward displacement is negative. Displacement is a vector; the signed values keep track of direction.
$\Delta x_1=+7.5\ \mathrm{m}, \Delta x_2=-2.0\ \mathrm{m}$
Add the displacements
For motion along one straight line, add the signed components to obtain the net displacement. The positive result points east.
$\Delta x_{\mathrm{net}}=\Delta x_1+\Delta x_2=+7.5\ \mathrm{m}-2.0\ \mathrm{m}=+5.5\ \mathrm{m}$
Represent the result
The numeric form is $+5.5\ \mathrm{m}$ . The vector diagram is a single arrow pointing east, labelled $5.5\ \mathrm{m}$ . The symbolic form shows the signed addition.

Answer: The net displacement is

5.5\ \mathrm{m}

east.

Check: Metres remain as the unit. The answer points east because the eastward part was larger. Its magnitude is less than 7.5 m, as expected after some westward motion.

Common mistakes and how to avoid them

Treating a negative position or velocity as a negative size.

Correction: A negative sign gives direction relative to the chosen origin or positive direction. State that convention before interpreting the sign.

Calling distance and displacement the same thing.

Correction: Distance is a scalar describing path length. Displacement is a vector describing the change from initial to final position. Use the quantity the question asks for.

Drawing an unlabeled graph or vector.

Correction: Label graph axes with quantities and units. Label vector arrows with the quantity, magnitude, and direction or sign convention.

Changing direction conventions partway through a solution.

Correction: Choose one positive direction and keep it for the calculation, graph interpretation, and vector diagram.

Lesson summary

Numeric, symbolic, graphical, and vector forms can describe the same physical situation.
Define the system, reference frame, and positive direction before using signed values.
A vector includes magnitude and direction; a scalar includes magnitude only.
For a straight position–time segment, velocity is the change in position divided by the change in time.
Check units, signs, graph labels, and arrow directions for agreement.

Check your understanding

Question 1

Right is positive. An object’s position changes from

+2.0\ \mathrm{m}

to

-4.0\ \mathrm{m}

in

3.0\ \mathrm{s}

. What is its average velocity?

$-2.0\ \mathrm{m/s}$
$+2.0\ \mathrm{m/s}$
$-6.0\ \mathrm{m/s}$
$+6.0\ \mathrm{m/s}$

Show answer and explanation

-2.0\ \mathrm{m/s}

The displacement is

-4.0\ \mathrm{m}-2.0\ \mathrm{m}=-6.0\ \mathrm{m}

. Dividing by

3.0\ \mathrm{s}

gives

-2.0\ \mathrm{m/s}

, so the motion is leftward.

Question 2

On a position–time graph, what does a horizontal segment mean for the object’s motion relative to the stated frame?

Its position is constant during that interval.
Its position is increasing at a constant rate.
Its velocity points in the positive direction.
Its displacement must be negative.

Show answer and explanation

Its position is constant during that interval.

A horizontal segment has no change in position as time passes. The object is at rest relative to the chosen frame during that interval.

Key terms

Scalar: A quantity with magnitude but no direction.
Vector: A quantity with both magnitude and direction.
Reference frame: The viewpoint or chosen surroundings used to describe position and motion.
Displacement: The change in position from an initial point to a final point, including direction.
Position–time graph: A graph showing an object’s position at different times.
Component: A signed part of a vector along a chosen direction or axis.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation A1.12. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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A1.12 · Use numeric, symbolic, graphical, and vector representations

Ontario Grade 12 Physics

Moving between numeric, symbolic, graphical, and vector representations

What you will learn

1. Begin with the system, frame, and direction

2. Four representations and what they show

3. Translate carefully and check for agreement

4. Build a consistent set of representations

From numbers to an equation and a graph

Read a position–time graph as a numeric relationship

Combine displacement vectors

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through SPH4U

About this lesson and its review