Apply the constant-acceleration kinematic equations to solve problems for displacement, initial and final velocity, acceleration and time, selecting the equation that omits the unknown not asked for.

What this topic is asking

This is the calculation core of Regents kinematics. The Physical Setting/Physics course gives you a set of constant-acceleration (uniform-acceleration) equations that link the five quantities of straight-line motion, displacement $d$, initial velocity $v_i$, final velocity $v_f$, acceleration $a$ and time $t$, and the exam asks you to pick the right one and solve. The four equations are printed on the Reference Tables, so the skill is selection and substitution, not memorizing.

The constant-acceleration equations

Each of the first three equations leaves out exactly one of the five quantities:

• $v_f = v_i + at$ contains no displacement.
• $d = v_i t + \tfrac{1}{2}at^2$ contains no final velocity.
• $v_f^2 = v_i^2 + 2ad$ contains no time.

That structure is the key to choosing quickly.

Choosing the right equation

For example, if a problem gives $v_i$, $a$ and $t$ and asks for $d$, the final velocity $v_f$ is not involved, so use $d = v_i t + \tfrac{1}{2}at^2$. If it gives $v_i$, $v_f$ and $d$ and asks for $a$, time is not involved, so use $v_f^2 = v_i^2 + 2ad$. Reading off which quantity is missing is faster than trying equations at random.

Sign conventions

Because velocity, displacement and acceleration are vectors in one dimension, choose a positive direction and apply it consistently. An object slowing down has an acceleration opposite to its motion, so $a$ is negative if motion is positive. Starting from rest means $v_i = 0$; coming to a stop means $v_f = 0$. Writing the knowns with their signs before substituting prevents most errors.

Multi-stage motion

If the acceleration changes (a car accelerates, then cruises at constant speed, then brakes), the equations apply to each stage separately, because each requires constant acceleration. Solve one stage, carry the final velocity and position forward as the initial values of the next, and add the displacements. A constant-velocity stage uses $d = vt$ (zero acceleration).

Reference Tables note

All the kinematic equations you need are printed in the Mechanics section of the Reference Tables, so you should never have to recall them. What you bring is the method: identify the missing quantity, choose the matching equation, and substitute with correct signs and units. The averaging relation $\bar{v} = \tfrac{1}{2}(v_i + v_f)$ is a quick check, since $d$ should equal the average velocity times the time.

Try this

Q1. An object accelerates from $4.0$ m/s at $2.0$ m/s squared for $3.0$ s. Calculate its final velocity. [2 points]

• Cue. $v_f = v_i + at = 4.0 + (2.0)(3.0) = 10.$ m/s.

Q2. State which kinematic equation you would use to find displacement when given $v_i$, $a$ and $t$ but not $v_f$. [1 point]

• Cue. $d = v_i t + \tfrac{1}{2}at^2$ (it omits the final velocity).