Topic 1.2: Displacement, Velocity, and Acceleration

The three definitions Topic 1.2 is built on

Everything in this topic follows from three definitions. Displacement is the change in an object's position:

Average velocity is the displacement of an object divided by the interval of time in which that displacement occurs:

Average acceleration is the change in velocity divided by the interval of time in which that change in velocity occurs:

Two features are worth noticing before any numbers arrive. All three are built from changes, so all three depend on a stated interval and a stated positive direction, not on a single instant. And none of the three is printed on the AP Physics 1 equation sheet, which starts at the constant-acceleration equations. These are definitions you carry in your head, and they hold whether or not the acceleration is constant.

The object model: why a car can be treated as a dot

Before you can say where something is, you have to decide what counts as the something. The CED answers with the object model: the size, shape, and internal configuration are ignored, and the object may be treated as a single point with extensive properties such as mass and charge.

That is a bigger simplification than it looks. A car has a bumper, a hood, and four wheels that all move slightly differently, and a sprinter's arm moves backwards while she runs forwards. The object model collapses all of that into one point carrying the car's mass or the sprinter's mass, so a single position value per axis describes the whole thing and the definitions above stay unambiguous. When a problem says a car travels 200 m, it means the representative point travels 200 m.

The model has limits, which is why the course later replaces the dot with a system and a center of mass in Unit 2, and with rotating rigid bodies in Unit 5. In Unit 1 the dot is always enough.

Displacement versus distance

Displacement depends only on where you start and where you finish. The definition $\Delta x = x - x_0$ contains no information about the route, so a hiker who walks a winding 8.0 km loop back to the car has a displacement of exactly zero while her distance is 8.0 km. Distance is the length of the path actually traveled, and it only accumulates: every extra step adds to it, whichever way that step points.

Three consequences follow, and they are worth stating explicitly.

• Distance is never smaller than the magnitude of displacement. They are equal only when the object moves in a straight line without reversing. A quarter circle of radius 100 m covers 157 m of path while displacing the object 141 m, with no reversal anywhere.
• Displacement can be negative or zero; distance cannot. A negative displacement means the finish is on the negative side of the start, which is direction information, not a smaller value.
• Displacement is path independent, so intermediate positions drop out. Given a messy route, you need only the first and last positions.

The sign convention that makes this work comes from Topic 1.1: choose a positive direction, then let the signs carry the directions.

Average velocity versus average speed

Average velocity divides displacement by the time interval, so it inherits every property of displacement, including the sign and the path independence. Its scalar counterpart, average speed, divides the total distance traveled by the same interval. The two agree only when the object moves in a straight line without reversing, and they can differ dramatically otherwise: the hiker who returns to her car has an average velocity of zero and an average speed of several kilometers per hour.

A second trap is subtler. Average velocity is not, in general, the average of the starting and ending velocities. Driving 100 km at 40 km/h and then 100 km at 60 km/h does not average to 50 km/h, because you spend more time in the slow half. The shortcut $v_{\text{avg}} = \frac{v_0 + v}{2}$ works only for constant acceleration, where equal time slices carry equal velocity increments. Under that condition it is a genuine time-saver, and it is one of the routes through the kinematic equations.

What accelerating actually means

An object is accelerating if the magnitude and/or direction of the object's velocity are changing. Read the and/or carefully, because it opens three doors rather than one.

The third row is the one worth rehearsing. A car going around a curve at an unchanging 15 m/s has a changing velocity, because velocity is a vector and its direction is turning, so it is accelerating even though the speedometer never moves. That is the whole basis of centripetal force in Unit 2. In one dimension, direction changes are limited to sign flips, which is why the pure direction-change case waits until you reach two dimensions.

Average versus instantaneous

Averages of velocity and acceleration are calculated considering the initial and final states of an object over an interval of time, so an average tells you nothing about what happened in between. Instantaneous values describe a single moment: your speedometer reads instantaneous speed, while a trip odometer and a clock give you an average.

The bridge between them is short. Calculating an average over a very small time interval yields a value that is very close to the instantaneous value. Take a ball dropped from rest, with the vertical axis pointing down so everything stays positive, which makes its position $y = 4.9t^2$ meters using $g = 9.8 \, \mathrm{m/s^2}$. Between 2.00 s and 2.10 s it falls $21.609 - 19.600 = 2.009$ m, giving an average velocity of $2.009 / 0.10 = 20.09 \, \mathrm{m/s}$. Shrink the interval to 2.00 s through 2.01 s and the average becomes $0.19649 / 0.010 = 19.65 \, \mathrm{m/s}$. The instantaneous velocity at 2.00 s is $(9.8)(2.00) = 19.6 \, \mathrm{m/s}$, and the averages close in on it as the interval shrinks. Topic 1.3 turns that squeeze into the slope of a tangent line, with no calculus required.

Speeding up or slowing down: read both signs

The CED flags one misconception by name in its Unit 1 overview: using negative acceleration exclusively to describe an object slowing down. The sign of the acceleration on its own settles nothing. Compare it with the sign of the velocity.

Same signs means speeding up, opposite signs means slowing down. One more case rounds it out: at the peak of a throw the velocity is momentarily zero while the acceleration is still $9.8 \, \mathrm{m/s^2}$ downward, so an object can be instantaneously at rest and accelerating hard at the same time. Zero velocity does not imply zero acceleration, and zero acceleration does not imply zero velocity.

How Topic 1.2 is tested, and what to do next

The CED lists four suggested skills for this topic: 1.C, create qualitative sketches of graphs that represent features of a model or the behavior of a physical system; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Those four sit beneath the three AP Physics 1 science practices, Creating Representations, Mathematical Routines, and Scientific Questioning and Argumentation. Unit 1 carries 10-15% of the multiple-choice section.

Read that skill list as a practice plan: sketch what the motion looks like, compute one unknown cleanly with units, then compare two moments or two scenarios and defend the comparison. From here, Topic 1.3 turns these definitions into graphs, the kinematic equations guide adds the constant-acceleration toolkit, the kinematics calculator checks your arithmetic, and the Unit 1 overview maps the rest of the unit.