Topic 1.2 Displacement, Velocity, and Acceleration: define displacement, velocity and acceleration as rates of change, and apply the kinematic equations to one-dimensional motion with constant acceleration.

What this topic is asking

The College Board (Topic 1.2) wants you to define displacement, velocity and acceleration as rates of change, to distinguish average from instantaneous quantities, and to apply the kinematic equations to one-dimensional motion where the acceleration is constant. This is the quantitative core of Unit 1: nearly every motion problem in the course reduces to choosing the right kinematic equation and substituting carefully.

Displacement, velocity and acceleration

Each quantity is the rate of change of the one before it: velocity tells you how fast position changes, and acceleration tells you how fast velocity changes. Because all three are vectors, each carries a sign in one dimension, and you must pick a positive direction before substituting numbers.

Average versus instantaneous

The distinction matters because real questions often give you start and end values (which set the average) but ask about a particular instant, or vice versa. When acceleration is constant the relationships are clean, which is exactly why the kinematic equations work.

The kinematic equations

For one-dimensional motion with constant acceleration, three equations relate displacement, the two velocities, acceleration and time:

Each equation omits one variable: the first has no displacement, the second has no final velocity, and the third has no time. The strategy is to list what you know, identify the unknown you want, and pick the equation containing only those quantities. These equations are valid only while the acceleration stays constant; if it changes (for example, a car that accelerates, then cruises, then brakes), you split the motion into segments and apply the equations to each in turn.

A useful companion relationship is that the displacement equals the average velocity times the time, $x - x_0 = \dfrac{v_0 + v}{2}\,t$, which follows from the definitions and is handy when you are not given the acceleration directly. On the AP exam the kinematic equations are printed on the equations sheet, so the marks come from selecting and applying them correctly, not from memorizing them.

Choosing signs and a frame

Because displacement, velocity and acceleration are all vectors, a problem can have, say, a positive velocity and a negative acceleration at the same time. That combination means the object moves forward while slowing down. The rule of thumb is that an object speeds up when velocity and acceleration share a sign and slows down when their signs differ. Free fall is the classic case: throw a ball up (positive velocity) under gravity (negative acceleration) and it slows, stops at the top, then speeds up downward, all with the same constant acceleration of $g \approx 9.8$ m/s squared directed down.

Try this

Q1. A ball is dropped from rest. Calculate its speed after $2.0$ s ($g = 9.8$ m/s squared). [2 points]

• Cue. $v = v_0 + at = 0 + (9.8)(2.0) = 19.6$ m/s downward.

Q2. A car travelling at $20$ m/s brakes at $-4.0$ m/s squared. Calculate how far it travels before stopping. [2 points]

• Cue. $v^2 = v_0^2 + 2ax \Rightarrow 0 = 20^2 + 2(-4.0)x \Rightarrow x = 50$ m.