Scalars and Vectors in Kinematics for HSC Physics

A car travels 60 m east, then turns around and travels 20 m west. How far did it travel? Easy: 80 m. But where did it actually end up? Only 40 m east of where it started.

Those two answers describe the same trip. The difference is that one quantity cares only about how much, while the other also cares about direction.

Before reading further, predict this: if the car drives 60 m east and then 60 m west, is its total motion zero?

Not quite. Its displacement is zero because it finishes where it started. Its distance travelled is 120 m. This distinction is the reason scalars and vectors matter in kinematics.

01The basic idea: some measurements need a direction

Imagine someone tells you, “The car is moving at 15 m/s.”

You know how fast it is moving, but you don’t know where it is going.

Now compare that with:

“The car is moving at 15 m/s east.”

That second statement contains more information. It gives both a size and a direction.

A scalar quantity has magnitude only. Magnitude means the size or numerical amount of the quantity.

A vector quantity has both magnitude and direction.

A useful first rule is:

If changing direction can change the quantity even when its size stays the same, the quantity is probably a vector.

For example, 10 m/s east and 10 m/s west have the same speed, but different velocities.

Check your understanding

A cyclist travels at 8 m/s north. What is the cyclist’s speed, and what is the cyclist’s velocity?

02Distance and displacement are not the same measurement

Distance asks:

How much ground did the object cover?

Displacement asks:

How far, and in what direction, did the object’s position change from start to finish?

Suppose you walk 30 m east and then 10 m west.

Your total distance is:

\[
30 + 10 = 40\text{ m}
\]

Your displacement is 20 m east because your final position is 20 m east of where you started.

The important point is that displacement does not care about every part of the journey. It compares only the initial and final positions.

A slightly silly way to picture it is this: distance is the friend who remembers every step of your complicated relationship history. Displacement ignores all of that and asks, “Fine, but where are you now compared with where you started?”

The analogy breaks because displacement is a precisely defined physical quantity, not emotional damage.

Practice question

A student walks 25 m north, then 7 m south. Find:

1. the total distance travelled
2. the displacement

03Direction can be represented using positive and negative signs

Writing “east” and “west” every time can become awkward. In one-dimensional motion, we usually choose one direction to be positive.

For example:

• east = positive
• west = negative

Then:

• \(+12\text{ m}\) means 12 m east
• \(-12\text{ m}\) means 12 m west

The choice is arbitrary. You could choose west as positive instead. Physics still works as long as you stay consistent.

This is called choosing a sign convention.

The negative sign does not automatically mean “slower”, “smaller”, or “bad”. It tells you the direction relative to the sign convention.

For example:

\[
v = -6\text{ m/s}
\]

could mean the object is travelling west at 6 m/s if east was chosen as positive.

Practice question

East is defined as positive. A car has a velocity of \(-18\text{ m/s}\).

What does the negative sign mean?

04Speed and velocity behave differently

Speed tells you how quickly distance is being covered.

Average speed is:

\[
\text{average speed} = \frac{\text{total distance}}{\text{total time}}
\]

Velocity tells you how quickly displacement is changing.

Average velocity is:

\[
v_{\text{av}} = \frac{\Delta x}{\Delta t}
\]

where:

• \(v_{\text{av}}\) is average velocity in metres per second, m/s
• \(\Delta x\) is displacement in metres, m
• \(\Delta t\) is the time interval in seconds, s

These formulas look almost identical. The important difference is the numerator.

Speed uses distance.

Velocity uses displacement.

05Returning to the starting point creates an important result

Suppose a runner completes one 400 m lap of a track and finishes exactly where they started.

Predict the average velocity.

You might be tempted to say it must be fairly large because the runner was moving quickly for the entire lap.

But the displacement is zero.

Therefore:

\[
v_{\text{av}}
= \frac{0}{\Delta t}
= 0\text{ m/s}
\]

The runner’s average velocity is zero, even though their average speed is not zero.

This does not mean the runner stood still. It means there was no net change in position.

That distinction appears regularly in HSC motion problems.

Practice question

A car travels 3 km east and then 3 km west in 10 minutes.

Find its displacement and average velocity.

06Vectors can cancel

Scalars usually add normally.

If you spend 4 s moving and then another 3 s moving, the total time is:

\[
4 + 3 = 7\text{ s}
\]

There is no direction attached to time.

Vectors are different because direction matters.

Suppose east is positive. A person moves:

\[
+12\text{ m}
\]

and then:

\[
-5\text{ m}
\]

Their total displacement is:

\[
\Delta x = 12 + (-5) = 7\text{ m}
\]

So the final displacement is 7 m east.

This is why opposite vectors can partially or completely cancel.

If the person instead moved \(+12\text{ m}\) and then \(-12\text{ m}\):

\[
\Delta x = 12 + (-12) = 0
\]

The distance travelled would still be 24 m.

07Acceleration is also a vector

Acceleration is often misunderstood as “speeding up”.

That is incomplete.

Acceleration describes the rate at which velocity changes.

\[
a = \frac{\Delta v}{\Delta t}
\]

where:

• \(a\) is acceleration in metres per second squared, m/s²
• \(\Delta v\) is the change in velocity in m/s
• \(\Delta t\) is the time interval in s

Because velocity is a vector, a change in velocity can happen when:

• the speed changes,
• the direction changes,
• or both change.

This means an object can accelerate even while its speed stays constant.

For example, a car travelling around a circular bend at a steady 15 m/s is accelerating because its direction, and therefore its velocity, is changing.

For straight-line HSC problems, the most important idea is that the signs of velocity and acceleration tell you their directions.

The common misconception: negative acceleration means slowing down

This is one of the most tempting mistakes in kinematics.

A negative acceleration does not automatically mean an object is slowing down.

It means the acceleration points in the negative direction.

To decide whether an object is speeding up or slowing down, compare the directions of velocity and acceleration.

The rule is:

If velocity and acceleration point in the same direction, the object speeds up. If they point in opposite directions, the object slows down.

Consider:

\[
v = -8\text{ m/s}
\]

and

\[
a = -2\text{ m/s}^2
\]

Both are negative, so both point in the same direction. The object is actually speeding up.

Its velocity might change from \(-8\text{ m/s}\) to \(-10\text{ m/s}\). The velocity becomes more negative, but the speed increases from 8 m/s to 10 m/s.

Practice question

East is positive. A car has:

\[
v = +20\text{ m/s}
\]

and

\[
a = -4\text{ m/s}^2
\]

Is the car speeding up or slowing down?

08A vector’s magnitude is always the size, not the signed value

Suppose:

\[
v = -12\text{ m/s}
\]

The velocity is \(-12\text{ m/s}\), but its magnitude is:

\[
|v| = 12\text{ m/s}
\]

The vertical bars mean “magnitude” or absolute value.

For velocity, the magnitude is the speed.

So:

\[
\text{speed} = |v|
\]

This distinction matters whenever a question asks for a magnitude.

A magnitude is not negative.

10Scalars and vectors on graphs

The scalar-vector distinction also helps when interpreting motion graphs.

On a displacement-time graph, displacement can be positive or negative because position is being measured relative to a chosen origin and direction.

The gradient of a displacement-time graph gives velocity:

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

A negative gradient therefore means negative velocity.

On a velocity-time graph, velocity can also be positive or negative.

A point below the time axis does not mean the object has “negative speed”. It means the object is moving in the negative direction.

The gradient of a velocity-time graph gives acceleration:

\[
a = \frac{\Delta v}{\Delta t}
\]

This is where the sign convention becomes especially useful. Instead of memorising separate rules for every graph shape, you can ask what direction the velocity and acceleration represent.

11What changes in two dimensions?

So far, we have mainly considered motion along one line. That lets us represent direction using positive and negative signs.

In two dimensions, a vector may need more information.

For example:

\[
20\text{ m/s at }30^\circ\text{ north of east}
\]

is a velocity vector.

It has:

• a magnitude of 20 m/s
• a direction of \(30^\circ\) north of east

You cannot fully describe that vector using only “positive” or “negative”.

Vectors in two dimensions can be broken into perpendicular components, usually horizontal and vertical components. This becomes important in projectile motion.

The scalar-vector idea itself does not change. Speed still gives only the magnitude of velocity, while velocity still requires magnitude and direction.

12A compact decision rule

When you meet a motion quantity, ask two questions.

1. Does direction form part of the physical meaning?

If no, it is a scalar.

Examples: distance, speed, time.

If yes, it is a vector.

Examples: displacement, velocity, acceleration.

2. If it is a vector, what sign convention or direction has been chosen?

Then keep that convention throughout the calculation.

This prevents many of the sign errors that appear later in kinematics equations.

13One final test

A car is travelling west and increasing its speed. East has been chosen as positive.

What signs should its velocity and acceleration have?