Unit 1 · Topic 1.1 Beta

Scalars and Vectors on a Line

Scalars have a size only; vectors have a size and a direction, which along a line is the sign.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

A fitness tracker says you walked 6 km today, yet you ended the day in your own kitchen, right where you started. Both statements are true. Physics keeps them apart: one number counts every step, the other only asks where you ended up, and the second one comes with a direction.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is 350 cm in meters?

  1. 3.5 m
  2. 35 m
  3. 0.35 m
  4. 3500 m
Show the answer

1 m = 100 cm, so 350 cm × (1 m / 100 cm) = 3.5 m.

  • Correct: 3.5 m:
  • 35 m:
  • 0.35 m:
  • 3500 m:

2. Which is the SI unit of time?

  1. second
  2. minute
  3. hour
  4. millisecond
Show the answer

The SI base units are the meter, the kilogram and the second.

  • Correct: second:
  • minute:
  • hour:
  • millisecond:

Part 4 · See it

See it first

A person starts at x = −1 m on a number line marked from −2 m to +6 m, walks 5 m in the positive direction to x = +4 m, turns around and walks 2 m back to end at x = +2 m. Dashed curved arrows show the two legs of the walk. A solid arrow from −1 m to +2 m is the displacement, +3 m. The distance traveled is 5 m + 2 m = 7 m, a scalar; the displacement is +3 m, a vector pointing in the positive direction.
A walk of 5 m out and 2 m back: the distance is 7 m, but the displacement is +3 m, from the start to the end. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. You pick an origin and a positive directionevery position along the line becomes one signed number, x, in meters
  2. An object moves from one position to anotherits displacement is the final position minus the starting position, with a sign for direction
  3. The object turns around along the waythe distance keeps growing, but parts of the displacement cancel, so the distance is larger
  4. Several displacements happen one after anothertheir vector sum is found by adding the signed values

Part 6 · Key ideas

Key ideas

  • A scalar has a size only (distance, speed, time). A vector has a size and a direction (position, displacement, velocity).
  • Choose a coordinate system: an origin and a positive direction. Then a position is one signed number, x.
  • Displacement = final position − starting position. Distance = the total length of the path. They match only for a trip in one direction.
  • Speed is how fast; velocity is how fast and which way. Along a line the sign of the velocity is the direction of motion.
  • Add vectors on a line by adding signed values: +320 m and −485 m give −165 m.

Part 7 · Misconception

A common mistake

The wrong idea: A negative velocity means the object is slowing down, and a negative position means it is moving backward.

What actually happens: The sign of a velocity only gives the direction of motion: −4.2 m/s means 4.2 m/s in the negative direction. A negative position only says which side of the origin the object is on.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

A walk along a hallway

Tape marks every meter along a straight hallway, with positive to the right. Mia starts at x = +2 m, walks to the 7 m mark (leg 1), turns around and walks to the 0 m mark (leg 2). The walk takes 20 s in all.

x (m)−101234567startleg 1leg 2end

Key: dashed arrows show the path of each leg; the faded figure marks the start.

1. What total distance does Mia walk?

Type a number in m.

Show the answer

Distance adds the length of every leg, with no signs: 5.0 m (from 2 m to 7 m) + 7.0 m (from 7 m to 0 m) = 12 m.

  • Answer: 12 m

2. What is Mia's displacement for the whole walk? Use a negative number for a displacement to the left.

Type a number in m.

Show the answer

Displacement is the final position minus the starting position: 0 m − (+2 m) = −2.0 m. The minus sign says it points left.

  • Answer: -2.0 m

3. How does the size of Mia's displacement compare with the distance she walks?

  1. It is equal, because both describe the same walk
  2. It is larger, because displacement includes a direction
  3. It is smaller, because displacement is measured in a different unit
  4. It is smaller, because she reversed direction partway
Show the answer

Distance is 12 m; the displacement is −2.0 m, size 2.0 m. Turning around makes the displacement smaller than the distance.

  • It is equal, because both describe the same walk: They match only for a trip in one direction. Mia turned around, so the two differ.
  • It is larger, because displacement includes a direction: A direction does not add size. The size of a displacement can never be more than the path length.
  • It is smaller, because displacement is measured in a different unit: Both are measured in meters; the difference comes from turning around, not from units.
  • Correct: It is smaller, because she reversed direction partway: Right: the 5 m out and most of the way back cancel in the displacement, but every meter counts toward distance.

4. During leg 2, what is the sign of Mia's velocity?

  1. Positive, because she is still at positive positions
  2. Positive, because speed is a positive number
  3. Negative, because she moves toward smaller x
  4. Zero, because she ends up back near the start
Show the answer

With right as positive, moving left means the velocity is negative, even while her position is still a positive number.

  • Positive, because she is still at positive positions: The sign of the velocity depends on the direction of motion, not on whether the position is positive.
  • Positive, because speed is a positive number: Speed is the size of the velocity and has no sign; the velocity has the sign of the direction.
  • Correct: Negative, because she moves toward smaller x: Right: her position decreases during leg 2, so her velocity points in the negative direction.
  • Zero, because she ends up back near the start: She is moving the whole time during leg 2; a zero velocity would mean she stands still.

5. A bus drives 235 m north along a straight street, then 410 m south. Taking north as positive, what is its total displacement?

Type a number in m.

Show the answer

Add the displacements with signs: +235 m + (−410 m) = −175 m, that is, 175 m south of the start.

  • Answer: -175 m

6. Two cyclists ride along the same straight road at 6.0 m/s, one heading east and one heading west. Which statement is correct?

  1. They have equal speeds and equal velocities
  2. They have equal speeds but different velocities
  3. They have different speeds but equal velocities
  4. They have different speeds and different velocities
Show the answer

Velocity is a vector: +6.0 m/s and −6.0 m/s (east positive) are different velocities with the same speed, 6.0 m/s.

  • They have equal speeds and equal velocities: Velocity includes direction; opposite directions mean different velocities.
  • Correct: They have equal speeds but different velocities: Right: speed has no direction, so it matches; velocity includes direction, so it differs.
  • They have different speeds but equal velocities: Both cover 6.0 m each second, so their speeds are equal, and their velocities point opposite ways.
  • They have different speeds and different velocities: Their speeds match: both are 6.0 m/s. Only the directions differ.

Part 9 · Summary

Summary

Scalars have a size only; vectors have a size and a direction, which along a line is the sign. Displacement is the final minus the starting position; distance is the path length. Speed is the magnitude of velocity, and vectors on a line add as signed numbers.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections