Other sections in this chapter introduced some of the fundamental equations of motion. These equations defined fundamental concepts; for example, average velocity equals the change in position divided by elapsed time.

Several other helpful equations can be derived from these basic equations. These equations enable you to predict an object’s motion without knowing all the details. In this section, we derive the formula shown in Equation 1, which is used to calculate an object’s final velocity when its initial velocity, acceleration and displacement are known, but not the elapsed time. If the elapsed time were known, then the final velocity could be calculated using the definition of velocity, but it is not.

This equation is valid when the acceleration is constant, an assumption that is used in many problems you will be posed.

Variables

We use t instead of Δt to indicate the elapsed time. This is simpler notation, and we will use it often.

Strategy

First, we will discuss our strategy for this derivation. That is, we will describe our overall plan of attack. These strategy points outline the major steps of the derivation.

1. We start with the definition of acceleration and rearrange it. It includes the terms for initial and final velocity, as well as elapsed time.
2. We derive another equation involving time that can be used to eliminate the time variable from the acceleration equation. The condition of constant acceleration will be crucial here.
3. We eliminate the time variable from the acceleration equation and simplify. This results in an equation that depends on other variables, but not time.

Physics principles and equations

Since the acceleration is constant, the velocity increases at a constant rate. This means the average velocity is the sum of the initial and final velocities divided by two.

We will use the definition of acceleration,

a = (v_f − v_i)/t

We will also use the definition of average velocity,

Step-by-step derivation

We have now accomplished our goal. We can calculate the final velocity of an object when we know its initial velocity, its acceleration and its displacement, but do not know the elapsed time. The derivation is finished.