AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/1/1-1)
Unit 1 · Topic 1.1
1.1 Scalars and Vectors in One Dimension
Every quantity in physics is either a scalar (just a size) or a vector (a size and a direction). In one dimension you handle direction with a plus or minus sign, which turns adding vectors into adding signed numbers. Getting signs right here prevents a huge share of mistakes later in the course.
Key terms
- scalar
- vector
- magnitude
- direction (sign convention)
- vector sum
- coordinate system
Scalars and vectors
A scalar is a quantity that only has a size, called its magnitude. Mass (3.0 kg), time (12 s), distance (40 m), speed (5 m/s) and energy (200 J) are all scalars. Asking "which way?" about them makes no sense.
A vector has a magnitude and a direction. Position, displacement, velocity, acceleration, force and momentum are all vectors. "5 m/s" is a speed; "5 m/s to the left" is a velocity.
You can draw a vector as an arrow. The arrow points in the vector's direction, and its length is proportional to its magnitude, so an arrow for 10 N is twice as long as an arrow for 5 N on the same diagram. In print, a vector symbol often has a small arrow over it, like v⃗.
Signs show direction in one dimension
When motion is along a single line, there are only two possible directions. You pick one as positive and the other becomes negative. That choice, plus an origin (where x = 0), is your coordinate system.
Once you've chosen, the sign of a component tells you its direction completely, so you don't need arrow notation. With right as positive, v = −4 m/s means 4 m/s to the left. With up as positive, a = −9.8 m/s² means 9.8 m/s² downward.
Any choice works, as long as you state it and stick with it for the whole problem. Up-positive is common for falling objects, but down-positive is fine for something that only falls.
Adding vectors along a line
To add one-dimensional vectors, add their signed values. Walking 5 m east and then 8 m west, with east positive, gives (+5 m) + (−8 m) = −3 m, so you end up 3 m west of where you started.
With arrows, place them tip to tail: draw the first arrow, start the second where the first one ends, and the sum (the resultant) runs from the start of the first to the tip of the last.
The magnitude of a vector is never negative. The vector −3 m has a magnitude of 3 m. A negative sign only says "pointing the negative way."
Signs inside equations
Equations like v = v₀ + at are really vector equations. In one dimension, every vector symbol in them stands for a signed number, so you plug in each value with its sign. With up as positive, a ball thrown upward at 8 m/s has v₀ = +8 m/s and a = −9.8 m/s², and one second later v = 8 + (−9.8)(1) = −1.8 m/s. The negative answer tells you the ball is already on its way down.
Scalars such as mass and time are never given signs. If an answer for a mass or a time comes out negative, you've made a sign error somewhere.
What a negative sign does not mean
A negative acceleration does not automatically mean slowing down. Whether an object speeds up or slows down depends on how the signs of velocity and acceleration compare:
| Velocity sign | Acceleration sign | What happens |
|---|---|---|
| + | + | speeds up, moving in the + direction |
| + | − | slows down, moving in the + direction |
| − | − | speeds up, moving in the − direction |
| − | + | slows down, moving in the − direction |
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Net displacement versus distance
A delivery robot moves along a hallway. It travels 4.0 m east, then 7.5 m west, then 2.0 m east. Taking east as positive, find (a) its net displacement and (b) the total distance it traveled.
Show the solutionHide the solution
- Step 1: Write each move as a signed number: +4.0 m, −7.5 m, +2.0 m.
- Step 2: (a) Displacement is the vector sum: (+4.0) + (−7.5) + (+2.0) = −1.5 m. The negative sign means the robot ends up west of its start.
- Step 3: (b) Distance is a scalar that adds up every bit of path, so use magnitudes: 4.0 + 7.5 + 2.0 = 13.5 m.
Answer: (a) −1.5 m, which is 1.5 m west; (b) 13.5 m
- Example 2Calculator allowed
Change in velocity when direction reverses (classic trap)
A ball is thrown straight up at 12 m/s. A moment later it is moving straight down at 5 m/s. Taking up as positive, find the change in the ball's velocity.
Show the solutionHide the solution
- Step 1: Give each velocity its sign: v_i = +12 m/s (up) and v_f = −5 m/s (down).
- Step 2: Δv = v_f − v_i = (−5) − (+12) = −17 m/s.
- Step 3: The trap is subtracting the speeds (12 − 5 = 7 m/s). That ignores the reversal. The ball lost 12 m/s of upward velocity and then gained 5 m/s of downward velocity, a 17 m/s change in all.
Answer: Δv = −17 m/s, which is 17 m/s downward
- Example 3Calculator allowed
Speeding up or slowing down?
With right as positive, a cart has velocity −3.0 m/s and acceleration −2.0 m/s². Is the cart speeding up or slowing down, and which way is it moving?
Show the solutionHide the solution
- Step 1: The velocity is negative, so the cart is moving to the left.
- Step 2: The acceleration has the same sign as the velocity, so it points the same way the cart moves.
- Step 3: When velocity and acceleration point the same way, speed increases, even though both numbers are negative.
Answer: It is moving left and speeding up.
Common mistakes
- Treating a negative acceleration as "slowing down." Compare the signs of velocity and acceleration: same sign means speeding up, opposite signs mean slowing down.
- Subtracting speeds instead of signed velocities when an object reverses direction. Write each velocity with its sign first, then compute v_f − v_i.
- Switching the positive direction halfway through a problem. Write "up is +" (or similar) at the start and use it for every vector.
- Calling distance or speed a vector. Both are scalars and are never negative; displacement and velocity are the vectors.
On the exam
- Multiple-choice questions often ask you to sort quantities into scalars and vectors, or to find a net displacement from several moves. Write each piece as a signed number before adding.
- In free response, state your sign convention ("taking up as positive"). Readers can then follow your signs, and a correct answer with a clear convention earns full credit even if your choice differs from theirs.
Connected topics
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Check yourself
4 questions on 1.1 Scalars and Vectors in One Dimension. Pick an answer to see if you got it, and why.
A student walks 30 m east, then 50 m west, then 10 m east along a straight hallway. Which gives the student's displacement and the total distance traveled?
Which of the following lists a scalar quantity followed by a vector quantity?
An object moves along a straight line. Which of the following quantities can never have a negative value, no matter which direction is chosen as positive?
Object 1 has a velocity of −8 m/s and object 2 has a velocity of +5 m/s. Which statement is correct?
0 of 4 answered