B3.1 Distinguish constant, instantaneous, and average motion quantities

Ontario Grade 11 Physics

Kinematics

SPH3U B3.1: Understanding what a motion value describes

A motion quantity can describe a whole trip, one moment, or an unchanging pattern. These meanings are different even when the same unit is used. This lesson focuses on distance, displacement, speed, and velocity. Distance and speed are scalars: they have magnitude, or size, but no direction. Displacement and velocity are vectors: they have both magnitude and direction. We will use one straight-line direction so that signs can show direction clearly.

What you will learn

Distinguish constant, instantaneous, and average motion quantities.
Tell scalar speed from vector velocity, including their units and direction.
Use displacement and elapsed time to calculate average velocity.
Interpret a motion graph without confusing an interval value with a single-moment value.

1. Prerequisite bridge: position, direction, and time

A system is the object whose motion is being described. In the examples, the system is a person or a cart. A reference point is a fixed place used to describe the system’s position. We describe position relative to that point.

Choose a positive direction before using signs. For example, let east be positive and west be negative. A position of

+4.0\,\mathrm{m}

is east of the reference point;

-4.0\,\mathrm{m}

is west. The sign indicates direction, while the number’s size gives the magnitude.

Elapsed time is the time between a start and an end. Use seconds, symbol

t

, as the SI unit of time. Position, distance, and displacement are measured in metres, symbol

\mathrm{m}

. Speed and velocity are measured in metres per second, symbol

\mathrm{m/s}

.

Distance is the total length of the path travelled. It is a scalar and cannot be negative. Displacement is the change in position from start to finish. It is a vector, so it includes direction. A person can travel a positive distance and still have zero displacement by returning to the starting point.

\Delta x=x_{\mathrm{f}}-x_{\mathrm{i}}

Set the system, reference point, and positive direction first.
Distance and speed are scalars; displacement and velocity are vectors.
Use metres and seconds, so motion rates are in metres per second.

2. Three ways to describe a motion quantity

A constant quantity keeps the same value over the stated time. Constant velocity means both the speed and direction remain unchanged. For example, a cart moving east at

2.0\,\mathrm{m/s}

without changing direction has constant velocity. Constant speed alone does not guarantee constant velocity if the direction changes.

An average quantity describes a complete time interval. Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time. Average velocity is a vector and can be positive, negative, or zero in our one-dimensional sign convention. Average speed is non-negative.

An instantaneous quantity describes one particular moment. Instantaneous speed is how fast an object is moving at that moment. Instantaneous velocity gives both the speed and the direction at that moment. A car’s speed display is intended to show its speed at that moment, not its average speed for the entire trip.

A motion graph can help separate these ideas. On a position-time graph, the horizontal axis shows time and the vertical axis shows position. A straight line with an unchanging slope represents constant velocity. The slope over a chosen interval represents average velocity for that interval; the slope of a straight segment also gives the velocity at any moment on that segment. If the graph bends, the motion is changing, and different parts of the graph can have different velocities. This lesson uses graph shape as a way to distinguish motion descriptions, not as a new calculation method.

The same trip may have different average and instantaneous values. A walker may pause, move quickly, and then slow down. One average velocity summarizes the net displacement over the full interval, while instantaneous velocity can differ from moment to moment. The average speed can also differ from the magnitude of average velocity because total distance and displacement are not always equal.

v_{\mathrm{avg}}=\frac{\Delta x}{\Delta t}

Constant means unchanged over the stated time.
Average describes an interval; instantaneous describes one moment.
Average speed uses total distance, while average velocity uses displacement.
A velocity needs a direction; a speed does not.

3. Choosing and interpreting the right quantity

Before calculating, ask what the question describes. If it asks for a whole interval, look for an average. If it asks for a single moment, it asks for an instantaneous quantity. If it says the value does not change, it describes a constant quantity. A constant value can also be the average over an interval, but the words answer different questions: one describes the motion pattern, and the other summarizes a time span.

For a one-dimensional trip, use a consistent positive direction. Find displacement by subtracting initial position from final position. Keep the sign when calculating average velocity. For average speed, use the total path length instead; do not attach a direction to the result.

A negative velocity does not mean the object has a negative speed. It means the object moves opposite to the chosen positive direction. A zero average velocity does not necessarily mean the object remained still: it may have moved away and returned, giving zero net displacement.

When reporting a result, include its unit and, for a vector, its direction or sign convention. Round the final value to a sensible number of significant figures based on the supplied values. Then check whether its size and direction fit the motion described.

v_{\mathrm{avg}}=\frac{x_{\mathrm{f}}-x_{\mathrm{i}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}

Match the requested type of quantity to the information provided.
Keep direction in displacement and velocity calculations.
Use total path length for average speed and net position change for average velocity.
A final answer should include units and a clear direction when needed.

Which motion quantity is being described?

Description	Meaning	Direction included?
Constant	Value stays unchanged over the stated time	For a vector quantity, yes
Average	One summary value for a time interval	For velocity, yes; for speed, no
Instantaneous	Value at one particular moment	For velocity, yes; for speed, no

Worked example

Average velocity on a straight trip

A student is the system. The reference point is the starting marker, and east is positive. The student moves from

x_{\mathrm{i}}=+2.0\,\mathrm{m}

to

x_{\mathrm{f}}=+14.0\,\mathrm{m}

in

6.0\,\mathrm{s}

. Find the average velocity.

Identify the quantity
The question asks for an average over the full time interval. The required vector quantity is average velocity. The known values are the initial and final positions and the elapsed time; the unknown is average velocity.
Find the displacement
Subtract the initial position from the final position. The positive result means the student’s net change in position is east.
$\Delta x=(+14.0\,\mathrm{m})-(+2.0\,\mathrm{m})=+12.0\,\mathrm{m}$
Calculate and report
Divide displacement by elapsed time. Keep the positive sign to show east, and report the result to two significant figures, matching the given measurements.
$v_{\mathrm{avg}}=\frac{+12.0\,\mathrm{m}}{6.0\,\mathrm{s}}=+2.0\,\mathrm{m/s}$

Answer: The student’s average velocity is

2.0\,\mathrm{m/s}

east.

Check: The units reduce to metres per second. The positive direction is east, matching the change in position. A

12.0\,\mathrm{m}

displacement over

6.0\,\mathrm{s}

gives a reasonable average of

2.0\,\mathrm{m/s}

.

Worked example

Average speed and average velocity can differ

A cart is the system. East is positive. It starts at

x=0.0\,\mathrm{m}

, travels

18.0\,\mathrm{m}

east, then

6.0\,\mathrm{m}

west. The trip takes

8.0\,\mathrm{s}

. Find its average speed and average velocity.

Find total distance
Average speed uses the full path length. Add the two parts of the trip. Distance is a scalar, so it has no direction.
$d_{\mathrm{total}}=18.0\,\mathrm{m}+6.0\,\mathrm{m}=24.0\,\mathrm{m}$
Calculate average speed
Divide total distance by elapsed time. This answer describes how much path the cart covers per second, on average.
$\text{average speed}=\frac{24.0\,\mathrm{m}}{8.0\,\mathrm{s}}=3.0\,\mathrm{m/s}$
Find displacement and average velocity
The cart ends $12.0\,\mathrm{m}$ east of where it started. Use this signed displacement, not total distance, to find average velocity.
$\Delta x=+18.0\,\mathrm{m}-6.0\,\mathrm{m}=+12.0\,\mathrm{m};\quad v_{\mathrm{avg}}=\frac{+12.0\,\mathrm{m}}{8.0\,\mathrm{s}}=+1.5\,\mathrm{m/s}$

Answer: Average speed is

3.0\,\mathrm{m/s}

. Average velocity is

1.5\,\mathrm{m/s}

east.

Check: Both results have units of metres per second. Average speed is greater because the cart travelled

24.0\,\mathrm{m}

but had only

12.0\,\mathrm{m}

of eastward displacement. The positive velocity direction agrees with the final position.

Worked example

Recognizing constant and instantaneous velocity

A cart moves east at an unchanged velocity of

-1.8\,\mathrm{m/s}

from

t=2.0\,\mathrm{s}

to

t=7.0\,\mathrm{s}

, with east defined as positive. State its instantaneous velocity at

t=5.0\,\mathrm{s}

and its average velocity over the stated interval.

Interpret the constant value
The statement says the velocity is unchanged throughout the interval. Therefore, its instantaneous velocity at the specified moment is the same as the constant velocity. The negative sign means the actual direction is west, so the wording “moves east” conflicts with the given sign.
Resolve the direction consistently
With east positive, a velocity of $-1.8\,\mathrm{m/s}$ points west. Use the sign convention and numeric value together; they cannot describe eastward motion here.
$v_{\mathrm{inst}}(5.0\,\mathrm{s})=-1.8\,\mathrm{m/s}$
Determine the interval average
Because the velocity remains constant for the entire interval, the average velocity over that interval is also the same value. The interval lasts $5.0\,\mathrm{s}$ , but no further calculation is needed.
$v_{\mathrm{avg}}=-1.8\,\mathrm{m/s}$

Answer: Using east as positive, both the instantaneous and average velocities are

1.8\,\mathrm{m/s}

west. The problem’s phrase “moves east” is inconsistent with its negative velocity.

Check: The unit is metres per second, and the negative sign indicates west under the stated convention. The instantaneous and average values match because the velocity is stated to be constant. This example also shows why direction words and signs must agree.

Common mistakes and how to avoid them

Calling average speed and average velocity the same quantity.

Correction: Average speed uses total distance and has no direction. Average velocity uses displacement and includes direction.

Dropping a negative sign from velocity without explaining the direction.

Correction: Keep the sign or state the direction in words. With east positive, a negative velocity points west.

Assuming zero average velocity means the object did not move.

Correction: An object can travel and return to its starting point. Its displacement, and therefore its average velocity, is then zero.

Treating an average value as the value at every moment.

Correction: An average summarizes an interval. It matches each instantaneous value only when the quantity stays constant throughout that interval.

Lesson summary

Constant describes an unchanged value over a stated time.
Average describes a whole interval; instantaneous describes one moment.
Average speed is total distance divided by elapsed time.
Average velocity is displacement divided by elapsed time and includes direction.
Use a consistent positive direction, include SI units, and check that the sign matches the stated motion.

Check your understanding

Question 1

A runner covers

40\,\mathrm{m}

in

10\,\mathrm{s}

. Which quantity can be found directly from these values?

Average speed, $4\,\mathrm{m/s}$
Instantaneous velocity at the finish, $4\,\mathrm{m/s}$
Constant velocity, $4\,\mathrm{m/s}$
Displacement, $4\,\mathrm{m/s}$

Show answer and explanation

Average speed,

4\,\mathrm{m/s}

Total distance divided by elapsed time gives average speed. The information does not establish the runner’s direction, instantaneous velocity, or constant motion.

Question 2

North is positive. An object has velocity

-3.0\,\mathrm{m/s}

. What is its direction?

North
South
It has no direction because velocity is a scalar
The direction cannot be identified even though the positive direction is known

Show answer and explanation

South

Velocity is a vector. A negative value points opposite to the chosen positive direction, so the object moves south.

Question 3

A cart’s velocity stays at

+2.5\,\mathrm{m/s}

for an interval. What is its average velocity over that interval?

$0\,\mathrm{m/s}$
$+2.5\,\mathrm{m/s}$
It must be greater than $+2.5\,\mathrm{m/s}$
It cannot be determined

Show answer and explanation

+2.5\,\mathrm{m/s}

When velocity is constant for the whole interval, the average velocity equals that constant value. The positive sign indicates the chosen positive direction.

Key terms

Scalar: A quantity with magnitude but no direction, such as distance or speed.
Vector: A quantity with magnitude and direction, such as displacement or velocity.
Displacement: The change in position from the starting point to the ending point.
Average velocity: Displacement divided by elapsed time for a chosen interval.
Instantaneous velocity: Velocity at one particular moment.
Constant velocity: Velocity that keeps the same magnitude and direction over the stated time.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation B3.1. 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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B3.1 · Distinguish constant, instantaneous, and average motion quantities

Ontario Grade 11 Physics

SPH3U B3.1: Understanding what a motion value describes

What you will learn

1. Prerequisite bridge: position, direction, and time

2. Three ways to describe a motion quantity

3. Choosing and interpreting the right quantity

Which motion quantity is being described?

Average velocity on a straight trip

Average speed and average velocity can differ

Recognizing constant and instantaneous velocity

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH3U

About this lesson and its review