Conceptual Physics

We have now developed two equations that involve time multiplied by a sum of velocities. The left side of the equation in step 11 matches an

expression appearing in equation 7, at the end of the first stage. By substituting from this equation into equation 7, we eliminate the time

variablet and derive the desired equation.

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.

Step Reason

12.vf^2 = vi^2 + a(2ǻx) substitute right side of 11 into 7

13.vf^2 = vi^2 + 2aǻx rearrange factors

2.15 - Motion equations for constant acceleration

The equations above can be derived from the

fundamental definitions of motion (equations such as

a =ǻv/ǻt). To understand the equations, you need

to remember the notation: ǻx for displacement, v for

velocity and a for acceleration. The subscripts i and f

represent initial and final values. We follow a common

convention here by using t for elapsed time instead of

ǻt. We show the equations above and below on the

right.

Note that to hold true these equations all require a

constant rate of acceleration. Analyzing motion with a

varying rate of acceleration is a more challenging

task. When we refer to acceleration in problems, we

mean a constant rate of acceleration unless we explicitly state otherwise.

To solve problems using motion equations like these, you look for an equation that

includes the values you know, and the one you are solving for. This means you can

solve for the unknown variable.

In the example problem to the right, you are asked to determine the acceleration

required to stop a car that is moving at 12 meters per second in a distance of 36

meters. In this problem, you know the initial velocity, the final velocity (stopped =

0.0 m/s) and the displacement. You do not know the elapsed time. The third motion

equation includes the two velocities, the acceleration, and the displacement, but does

not include the time. Since this equation includes only one value you do not know, it is

the appropriate equation to choose.

ǻx = displacement, v = velocity, a = acceleration, t = elapsed time

Applying motion equations

Determine the “knowns” and the

“unknown(s)”

Choose an equation with those

variables

·Find other knowns from situation

Motion equations

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