AP PHYSICS C: MECHANICS › UNIT 1, KINEMATICS › TOPIC 1.1
Scalars and Vectors in One Dimension
This is the shortest topic in the course and the one that quietly decides the most. Almost every lost point in Unit 1 is a sign, and almost every sign traces back to a coordinate system that was never written down.
1.1.A Tell a scalar from a vector, and say what each one carries.
1.1.B Describe a vector quantity relative to a coordinate system you chose yourself, and know which parts of the description your choice can change.
Wording here is mine, written from the Day 1 and Day 2 Cornell notes. Check it against your own copy of the CED.
1. The coordinate system is a choice, and you have to make it
A scalar is a number with a unit. Mass, time, speed, kinetic energy, temperature, charge. It is complete on its own. Nothing about it depends on which way you decided was forward.
A vector carries a magnitude and a direction, and that direction has to be measured against something. There is no natural something. You supply it. The moment you write +x on a diagram you have made a physical claim about how you are going to read every number that follows, and every sign in the problem now answers to it.
In one dimension there are only two directions, so the choice collapses to a sign. A vector along the axis is fully described by one signed number, and physicists write that number with the unit vector attached:
r = x î,where î points along +x and has length 1, no units
The î is not decoration. It is the piece that remembers the axis. Write x = −26 m and a reader has to hunt for your diagram; write r = −26 î m and you have said which way you are measuring in the same breath as how far.
One vector, one axis, two ways to name the same thing
Drag the head of the arrow. The drone really is somewhere, and dragging moves it. Then press the button and rename the directions without moving anything at all.
Two numbers describe that arrow. One of them changed when you flipped the axis and one of them did not, and the difference between them is the entire content of this topic.
2. Magnitude, component, and the absolute value that is not one
For a one dimensional vector the magnitude is the absolute value of the component:
|r| = |x| ≥ 0,always
Two consequences worth saying out loud, because both show up on free response papers every year.
A magnitude is never negative. If a question asks for the magnitude of an acceleration and your answer carries a minus sign, you have answered a different question. Report 9.8 m/s2, directed downward, or report a = −9.8 ĵ m/s2 with ĵ up. Do not report a negative magnitude.
Distance is not the magnitude of displacement. It is the accumulated path length, and it only equals |Δx| when the motion never reverses. A car that drives 300 m east and 300 m back has covered 600 m and is displaced by nothing. You will meet the general version in Topic 1.2 as an integral of speed rather than of velocity.
3. Which quantities are which
Sorting them is not memorization. Ask the invariance question: if I turned my axis around, would this number change sign?
Pick a quantity, then say what it is. Wrong answers explain themselves rather than just going red.
Two of those are worth arguing about. Speed is the magnitude of velocity, so it is a scalar even though velocity is not, which is why a speedometer can be honest while telling you nothing about where you are going. Current has a direction drawn on every circuit diagram you will ever see and is still a scalar, because that direction is a property of the wire and not of the quantity: current does not add head to tail, it adds at a junction.
4. Looking ahead, and the quadrant your calculator will not give you
Topic 1.5 does this properly. It is here because your summer packet already asked for it at item 18, and because the trap is worth meeting twice.
In a plane a vector is two components, and the two descriptions convert both ways:
A = Ax î + Ay ĵ|A| = √(Ax2 + Ay2)tan θ = Ay / Ax
That last equation is true and its inverse is not. The tangent function repeats every 180 degrees, so arctan on your calculator has to pick one of two answers, and it always picks the one between −90° and +90°. For anything pointing left, that is the wrong one by exactly half a turn.
Drag the head of the vector anywhere in the plane. The gold arrow is what your calculator returns from arctan alone. The blue arrow is where the vector actually points. Watch what happens when you cross into the left half.
Check yourself
1. A drone flies 26 m east. Write its displacement in unit vector form with east positive, then again with west positive, and say which of the two numbers you reported is a fact about the drone.
East positive: r = +26 î m. West positive: r = −26 î m. In both cases |r| = 26 m. The magnitude is the fact about the drone. The sign is a fact about your page.
2. A student writes that a ball in free fall has acceleration of magnitude −9.8 m/s2. Name the error, and give two correct ways to say what the student meant.
A magnitude is a length and cannot be negative. Either a = −9.8 ĵ m/s2 with ĵ upward, or a magnitude of 9.8 m/s2 directed downward. The minus sign belongs to the component, and it exists only because the student chose up as positive.
3. A vector has components Ax = −7 and Ay = 24. Find its magnitude and its direction, and state what your calculator returns for arctan(24 / −7).
|A| = √(49 + 576) = √625 = 25. The calculator returns −73.7°, which points down and to the right. The vector points up and to the left, in the second quadrant, so the direction is −73.7° + 180° = 106.3° measured counterclockwise from +x. A sketch would have told you that before the calculator did.
4. A car drives 300 m east, then 300 m west, in 40 s. Give the distance, the magnitude of the displacement, the average speed and the average velocity.
Distance 600 m, displacement magnitude 0. Average speed 600 / 40 = 15 m/s, average velocity 0. The pair of zeros is the point: the car was moving the entire time, and average velocity is built only from the endpoints.
Topic 1.2, Displacement, Velocity, and Acceleration. The same quantities with a clock on them, and the point where this course parts company with AP Physics 1: the slope becomes a derivative and the area becomes an integral.