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B. Kinematics · Kinematics: Motion in One and Two Dimensions

Describing Motion: Scalars, Vectors, Position and Displacement

B2.1 B3.2 A1.12 A1.13About 70 minutes
Learning goals
  • I can explain what kinematics studies and how it differs from dynamics.
  • I can tell scalar quantities from vector quantities and give examples of each.
  • I can describe position relative to a reference point using a sign convention or a direction in square brackets.
  • I can calculate distance and displacement for motion along a straight line.
  • I can report measured and calculated values with an appropriate number of significant digits.

Words to know

kinematics
The study of how objects move (position, velocity, acceleration, time) without asking what causes the motion.
dynamics
The study of why objects move the way they do: forces and their effects. You study it in the Forces unit.
scalar quantity
A quantity that has magnitude (size) only, such as time, distance, speed, mass and temperature.
vector quantity
A quantity that has both magnitude and direction, such as position, displacement, velocity, acceleration and force.
reference point (origin)
The point you measure positions from. Its position is zero.
position (d)
The distance and direction of an object from the reference point, for example 12 m [E].
distance (d)
The total length of the path travelled. A scalar; it is never negative.
displacement (Δd)
The change in position: Δd = d₂ − d₁. A vector that points from the starting position to the final position.
time interval (Δt)
The amount of time between two instants: Δt = t₂ − t₁.
significant digits
The digits in a measurement that are known with certainty plus one estimated digit.

Where did you end up?

Picture a hockey practice on an outdoor rink. Your coach blows the whistle and you skate from the goal line to the far boards, 60 m away, and then straight back. You are out of breath, so you clearly went somewhere: you skated 120 m. But if someone asks, "How far are you from where you started?" the honest answer is zero.

Both answers are correct. They describe two different things. Physics gives each one its own name, distance and displacement, and the difference between them is the first big idea of this course.

Remember from Grades 9 and 10

In Grade 9 math you found the rate of change of a linear relation and connected it to the slope of its graph. In Grade 9 science you calculated speed as change in distance ÷ time. In Grade 10 math you used trigonometry (sine, cosine, tangent, the sine law and the cosine law) and solved quadratic equations, including with the quadratic formula. You will use every one of these tools in this unit.

What is kinematics?

Kinematics is the part of physics that describes motion. It answers questions like: Where is the object? How fast is it going, and in which direction? Is it speeding up? Kinematics does not ask why an object moves. That question belongs to dynamics, which you study in the Forces unit.

To describe motion clearly, physicists use a small set of quantities: time, distance, position, displacement, speed, velocity and acceleration. This lesson starts with the first four.

Scalars and vectors

Some quantities need only a number and a unit. These are scalars. "The game lasted 60 min" and "the water is 4 °C" are complete descriptions.

Other quantities also need a direction. These are vectors. "The canoe moved 300 m" is incomplete if you are trying to find the canoe. "The canoe moved 300 m [N]" tells you exactly where to look.

  • Scalars: time, distance, speed, mass, temperature, energy.
  • Vectors: position, displacement, velocity, acceleration, force.

In this course we write a vector's direction in square brackets after its magnitude: 25 m [S], 8.0 m/s [W], 3.2 m/s² [up]. Textbooks also put a small arrow over the symbol, like d⃗, to show it is a vector. We draw a vector as an arrow: its length shows the magnitude (using a scale) and it points in the vector's direction.

Scale: 1 square = 10 mA = 30 m [E]B = 20 m [W]C = 30 m [W]A and C: samemagnitude,oppositedirections,so A ≠ C
Vectors drawn to scale. Each arrow's length is its magnitude and its arrowhead shows its direction. A and C have the same magnitude (30 m) but are different vectors, because their directions differ.

Position and the sign convention

To give a position you need a reference point (also called the origin). Position is the distance and direction from that point. If the reference point is your front door, a mailbox 15 m east has position d = 15 m [E].

For motion along one straight line, it is often easier to use + and − signs instead of compass words. You choose one direction to be positive and state it: "Let east be positive." Then

  • 15 m [E] is written +15 m, and
  • 15 m [W] is written −15 m.

The sign is not about "good" or "bad". It only tells direction. You can choose either direction as positive, as long as you say which one and stick with it for the whole problem.

Distance and displacement

Distance is the total length of the path you travelled. It is a scalar, so it has no direction and is never negative.

Displacement is the change in position:

Δd = d₂ − d₁

where d₁ is the initial position and d₂ is the final position. (Δ is the Greek capital letter delta and means "change in": final minus initial.) Displacement is a vector that points straight from where you started to where you finished. The path in between does not matter.

start: homeend: schooldisplacement = 1.2 km [E] (straight arrow)distance = 1.8 km(length of the dashed path)NE →
The walk to school follows the winding dashed path, so the distance is 1.8 km. The displacement is the straight arrow from start to finish: 1.2 km [E]. (Not to scale.)
Worked example

Example 1: Walking the dog

Lena leaves her front door and walks her dog 50 m [E] along the sidewalk. The dog spots a squirrel, so they turn around and walk 70 m [W]. Find (a) the distance they travelled, (b) their final position, and (c) their displacement.

  1. Choose the front door as the reference point and let east be positive.
  2. (a) Distance adds the lengths of the path: 50 m + 70 m = 120 m. No direction is needed.
  3. (b) Start at 0. Moving 50 m [E] gives +50 m. Moving 70 m [W] adds −70 m: 50 + (−70) = −20 m. The final position is 20 m [W] of the door.
  4. (c) Δd = d₂ − d₁ = (−20 m) − 0 = −20 m, so the displacement is 20 m [W].

Distance = 120 m; final position = 20 m [W]; displacement = 20 m [W].

-30-20-100102030405060+50 m−70 m
Example 1 on a number line (metres, east positive). The two jumps add up to a distance of 120 m, but the walk ends at −20 m, so the displacement is 20 m [W].
Worked example

Example 2: Displacement between two positions

A delivery truck is at position 12 km [W] of the town hall at 9:00 a.m. At 9:30 a.m. it is at 30 km [E] of the town hall. Find its displacement.

  1. Let east be positive. Then d₁ = −12 km and d₂ = +30 km.
  2. Δd = d₂ − d₁ = (+30 km) − (−12 km) = 30 + 12 = +42 km.
  3. The positive sign means east, so Δd = 42 km [E].
  4. Check: going from 12 km west of the town hall to 30 km east of it means passing the town hall, so 12 + 30 = 42 km. ✓

Δd = 42 km [E].

Time and time intervals

Physics also separates an instant from an interval. A time t is a reading on a clock or stopwatch, such as t = 3.0 s. A time interval Δt = t₂ − t₁ is how long something took. If a stopwatch reads 3.0 s when a runner passes one cone and 7.5 s at the next cone, the time interval is Δt = 7.5 s − 3.0 s = 4.5 s. Time intervals in kinematics are always positive.

Significant digits (you will use these all year)

A measurement is only as precise as the tool that made it. Significant digits are the certain digits plus one estimated digit.

  • Non-zero digits are significant: 4.73 m has 3.
  • Zeros between them are significant: 1.05 s has 3.
  • Leading zeros are not: 0.0042 kg has 2.
  • Trailing zeros after a decimal point are significant: 2.50 m has 3.
  • Trailing zeros in a whole number are unclear (does 300 m have 1, 2 or 3?). Use scientific notation: 3.00 × 10² m has 3.

Multiplying or dividing: round the answer to the fewest significant digits in the data. Adding or subtracting: round to the fewest decimal places in the data. Keep extra digits during the steps and round only the final answer. Exact counts (like "2 laps") do not limit significant digits.

Worked example

Example 3: Rounding a result

(a) A runner's positions are measured as 12.4 m and 3.15 m. Find the displacement between them (east positive, both east of the origin, runner moving west). (b) A car travels 125 m in 4.2 s. Write its average speed with the correct number of significant digits.

  1. (a) Δd = d₂ − d₁ = 3.15 m − 12.4 m = −9.25 m. This is a subtraction, so round to the fewest decimal places: 12.4 has one decimal place. Δd = −9.3 m, or 9.3 m [W]. (−9.25 rounds to −9.3 by the usual round-half-up rule.)
  2. (b) Speed = distance ÷ time = 125 m ÷ 4.2 s = 29.761... m/s. This is a division: 125 has 3 significant digits and 4.2 has 2, so the answer gets 2.
  3. Speed ≈ 30 m/s. Written this way the zero is meant to count, so it is clearer to write 3.0 × 10¹ m/s.

(a) 9.3 m [W]; (b) 3.0 × 10¹ m/s (30 m/s).

Just for fun

Why did the vector break up with the scalar?

It wanted a relationship with some direction.

Hands-on activity

Build a hallway number line

You need: masking tape, a measuring tape or metre stick, a marker, and a partner.

  1. Stick a long strip of tape on the floor of a hallway (or a driveway, using chalk outdoors). Mark a reference point 0 near the middle, and mark every 0.5 m in both directions. Label one direction "+".
  2. Your partner reads instructions such as "walk to +2.5 m, then to −1.0 m, then to +1.5 m". Walk them.
  3. For each trip, record the starting and final positions. Calculate the distance for the whole walk and the displacement (Δd = d₂ − d₁).
  4. Make up a walk whose displacement is zero but whose distance is more than 6 m.

What to look for: for the sample walk (start at 0), the distance is 2.5 + 3.5 + 2.5 = 8.5 m, and the displacement is +1.5 − 0 = +1.5 m. A zero-displacement walk must end where it started (for example 0 → +3.5 → 0, which is 7.0 m of walking).

Your turn

1

Sort these into scalars and vectors: 15 s, 30 m [N], 72 km/h, 9.8 m/s² [down], 4.0 kg, 12 m [W].

2

A swimmer swims 4 lengths of a 25 m pool and ends where she started. Find her distance and displacement.

3

With north positive, write 18 m [S] using a sign.

4

A cyclist moves from 40 m [E] to 15 m [E] of a stop sign. Find the displacement.

5

A bus moves from 3.0 km [S] of a school to 5.0 km [N] of it in a straight line. Find its displacement.

6

How many significant digits are in (a) 0.0560 m, (b) 2.07 × 10³ s, (c) 100.0 g?

7

Can distance ever be less than the magnitude of displacement? Explain.

More practice: warm-up

1

What does the symbol Δ mean?

2

Is speed a scalar or a vector?

3

A stopwatch reads 2.0 s, then 9.5 s. What is the time interval?

4

You walk 6 m [N] and then 6 m [S]. What is your displacement?

5

Round 0.04567 m to 2 significant digits.

More practice: on your own

1

A snowplow drives 2.4 km [E], then 3.1 km [W], then 0.9 km [E]. Find the distance and displacement.

2

An elevator goes from the 8th floor down to the ground floor, then up to the 3rd floor. Each floor is 3.5 m apart (ground = floor 0). Find the distance and displacement.

3

Calculate and round correctly: 4.52 m + 12.1 m + 0.383 m.

4

Calculate and round correctly: 48.0 m ÷ 3.2 s.

5

Explain why "the train's position is 20 km" is an incomplete description.

Challenge

1

A runner goes around a circular track of radius 50.0 m. After half a lap, what are her distance and the magnitude of her displacement?

2

A ball rolls from position −2.5 m to +4.0 m, then back to +1.0 m. Find the distance, the displacement for the whole trip, and the displacement for the second part only.

3

Two students measure a hallway. Ari reports 12 m and Bea reports 12.00 m. Are they saying the same thing? Which one needed the better measuring tool?

Check your understanding

Quiz

1.

Which list contains only vector quantities?

2.

A dog runs 30 m [N] and then 10 m [S]. Its displacement is:

3.

A car moves from 8.0 km [W] to 3.0 km [E] of a gas station. With east positive, Δd is:

4.

Which statement about distance and displacement is always true?

5.

How many significant digits does 0.00340 m have?

6.

Kinematics is the study of:

For parents and teachers

Big idea: This lesson sets up the language for the whole unit (expectations B2.1 and B3.2): scalars versus vectors, position, distance, displacement and time interval. It also introduces the significant-digit rules (A1.13) and vector arrows (A1.12) that students use all year.

How to help: Use everyday trips. "We drove to the grocery store and back: what was our distance? Our displacement?" Ask your teen to always say which direction is positive before using signs.

Common difficulties:

  • Treating a negative displacement as "wrong". The sign only shows direction.
  • Computing d₁ − d₂ instead of d₂ − d₁. Displacement is always final minus initial.
  • Adding displacements as if they were distances (ignoring direction).
  • Rounding in the middle of a calculation, or ignoring significant digits entirely. Encourage keeping all calculator digits until the final answer.