## Wednesday, January 25, 2017

### Concepts of Acceleration

Before starting an exploration of acceleration, it is useful to revisit how we think about velocity. Previously, velocity was described like this:

average velocity: the value represented by the displacement of an object in relationship to the elapsed time. Remember that displacement is the measure of the length of a line lying directly between the initial position and a final position of an object that moved.

What is key in this is that velocity is the rate at which the position of an object changes. The equation that relates displacement to velocity and time is:

d = vt

where d is displacement, v is average velocity, and t is elapsed time.

To complicate things (because that's what we do) displacement can be re-imaged as a change of position—the difference between initial position and final position—and elapsed time can be re-imaged as the difference between initial time and final time.

Okay, so if complicated formulas intimidate you, it's okay to close your eyes for a few seconds!

The re-imaged equation becomes, where:

di = initial position (or beginning displacement from some reference point)
df = final position (or final displacement from some reference point)
ti = initial time (clock reading at the beginning, when the object is at di)
tf = final time (clock reading at the end, when the object is at df)
v = average velocity

(d - di) = v ( ti - ti)

Okay, you can open your eyes now!

The "ah ha moment" above is that velocity relates to how the position of the object changes. Using words instead of math symbols, the idea is a lot more logical.

Velocity is the rate that position changes over some period of time.

This could be stated another way with displacement being the subject. Think about it this way:

The change in position of an object is determined by its rate of motion (velocity) and how long it moves.

Further, if it is some distance from a reference point to begin with (di), it will end up that distance plus how far it moved. That is evident in this equation:

df = di + vt

(back to that, yeah…)

Bam!

Say that about 10 times!

On To Acceleration!

Now, with the understanding of velocity refined, understanding acceleration becomes much simpler. There is a perfect parallel between acceleration concepts and velocity concepts!

To begin, imagine sitting in a car at a red light. When the light turns green, the driver presses the gas pedal. WHICH is CALLED the accelerator because… Pressing the accelerator, the flow of fuel to the engine increases, the RPMs increase. The passengers of the car feel themselves pressed back into the seat as the car moves forward. The speedometer begins to show a change in speed

5 MPH
10 MPH
20 MPH
40 MPH

That image is more than is needed to understand the concept of acceleration!

Acceleration is the rate that velocity changes over some period of time.

The parallels between acceleration and velocity carry through to the math equations as well. Just to prove the point, the previous discussion of velocity is going to be copied below this sentence and all the relevant word changes will be made in red!

What is key in this is that acceleration is the rate at which the velocity of an object changes. The equation that relates velocity to acceleration and time is:

vf = vi + at

where v is change in velocity, a is rate of acceleration, and t is elapsed time.

To complicate things (because that's what we do) change in velocity can be re-imaged as the difference between initial velocity and final velocity, and elapsed time can be re-imaged as the difference between initial time and final time.

Okay, so if complicated formulas intimidate you, it's okay to close your eyes for a few seconds!

The re-imaged equation becomes, where:

vi = initial velocity (or beginning velocity from some reference point)
vf = final velocity (or final velocity from some reference point)
ti = initial time (clock reading at the beginning, when the object is at di)
tf = final time (clock reading at the end, when the object is at df)
a = acceleration rate

(v - vi) = a ( ti - ti)

Okay, you can open your eyes now!

The "ah ha moment" above is that acceleration relates to how the velocity of the object changes. Using words instead of math symbols, the idea is a lot more logical.

Acceleration is the rate that velocity changes over some period of time.

This could be stated another way with change in velocity being the subject. Think about it this way:

The change in velocity of an object is determined by its rate of acceleration and how long it moves. Or, to save pixels:

v = at

(back to that, yeah…)

So you can see that acceleration and velocity are similar, but that velocity relates to the change of position and acceleration relates to the change of velocity.

Another parallel concept between acceleration and velocity is when the object has an initial velocity and then undergoes acceleration. This is just like, with displacement, when an object is some distance from a starting point and begins to move. For velocity, the equation is:

df = di + vt

where df is the final distance from a reference point, di is the initial distance from a reference point, v is average velocity and t is elapsed time.

The relationship between final velocity and acceleration is found using a parallel equation:

vf = vi + at

where vf is the final velocity, vi is the initial velocity, a is acceleration rate and t is elapsed time.

In many cases, the "final velocity" concept is easier to visualize than is "final distance." Look at the two examples that follow.

Final Distance
A car starts out 4 miles from the state line and travels at an average velocity of 20 miles per hour for 2 hours. How far from the state line does it end up?

For the above, the equation is (how far does it end up?):

df = di + vt

df = 4 mi + 20mi/hr • 2 hr

df = 4 mi + 40mi
df = 44 mi

While this is not terribly complicated, processing the idea of "how far from the state line" is a bit unfamiliar. The idea that the car moves 20 MPH for 2 hours is fairly simple—40 miles is traveled. So, the car moves 40 miles from the initial position, which is 4 miles from the state line, so… Well, in the end, it seems a little less obvious.

HOWEVER, with acceleration, the idea of final velocity is more readily accessible. The basis of the idea is that something is moving at a given velocity ( ) and either speeds up or slows down. Thus, it has a new, different velocity. Hence:

vf = vi + at

Final Velocity
A car starts out traveling at 40 mi/hr. It speeds up at a constant rate of 10 mph/sec for 2 second. How fast does it end up traveling?

For the above, the equation is (How fast does it end up traveling?):

vf = vi + at
vf = 40 mi/hr + 10 (mi/hr)/sec • 2 sec

vf = 40 mi/hr + 20 mi/hr

So the change in speed is + 20 mi/hr. Thus, the final velocity is…

vf = 60 mi/hr

In many cases, the initial velocity will be zero. This will be true whenever the situation includes an object "at rest" or "not moving." Here are some phrases to look for that allow you to assume the initial velocity is zero:

An object is at rest…Two stationary zebras…The airplane begins to move…

Generally, unless there is some definite indication of an initial velocity, you can assume vi = 0 and if so, the equation reduces nicely to:

vf = at

However, it is good practice to start with the complete equation each time, fill in the give values, and then solve. One more step is not going to kill you!