Displacement and Acceleration

If the relationship between displacement, acceleration, and elapsed time is extended to the most general case, the following equations need to be combined.

1.)

df = di + vt

where df is the final displacement from the fixed point, di is the initial displacement from the fixed point, v is rate of change in position, (average velocity), and t is elapsed time.

2.)

Vf = Vi + at

where Vf is final velocity, Vi is initial velocity, a is the rate of acceleration, and t is the elapsed time.

3.)

Vave = (Vf  + Vi) / 2

where Vf is final velocity, Vi is initial velocity, and Vave is the average velocity.

To begin, start with the general displacement equation:

df = di + vt

Understanding that v is average velocity, the equation becomes:

df = di + ((Vf  + Vi) / 2) • t

And since Vf can be found in relationship to acceleration, the following emerges:

df = di + (((Vi + at) + Vi) / 2) • t

Solving the problem in steps generally is more easily understood. To find displacement when acceleration is present, do the following.

Step 1.) Find the final velocity: Vf = Vi + at

Step 2.) Find the average velocity: Vave = (Vf  + Vi) / 2

Step 3.) Find the displacement: df = di + vt

EXAMPLE

A model car is 3 meters from the starting line on a model car race track and it is moving at 2 m/s. It accelerates at a rate of 5 (m/s)/s for 4 seconds. How far does it end up from the starting line?

Step 0.) Collect the data!

di = 3 m

vi = 2 m/s

a = 5 (m/s)/s

t = 4

Step 1.) Find final velocity.

Vf = vi + at

vf = 2 + 5•4

vf = 2 + 20

vf = 22

Step 2.) Find average velocity.

v(ave) = (vf + vi)/2

v(ave) = (22 + 2)/2

v(ave) = 24/2

v(ave) = 12 m/s

Step 3.) Find the total displacement.

df = di + vt

(remember v is average velocity and t is the same elapsed time as for the acceleration)

df = 3 + 12•4

df = 3 + 48

df = 51 meters 

So, the model car ends up 51 meters from the starting line.

Acceleration, Final Velocity, and Average Velocity

What is acceleration? Most people more or less know this. But, let's start with an example!

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 causing the RPMs of the engine to 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 effect of acceleration is to change velocity. This can be explored mathematically quite easily.

EXAMPLE:

Suppose a car is rolling along a highway at 55 MPH and the speed limit changes to 65. How much did it accelerate (speed up)?

Inspection of the numbers answers that question. It speed up by 10 MPH.

So… using vf for final velocity, vi for initial velocity, and ∆d for how much velocity changed:

vf = 65 mph

vi = 55 mph

∆v = 10 mph

The equation is equally intuitive:

vf = vi + ∆v

Suppose we wanted to find vf?

Find vf where
vi = 55 mph
∆d = 10 mph

vf = vi + ∆v

vf = 55 mph + 10 mph
vf = 65 mph

(Of course, you could solve for any of the numbers!)

There is a strong similarity between changing velocity and changing position.

When an object changes position, the rate of that change is velocity. When an object changes velocity, the rate of that change is acceleration. It is a whole lot like what has been learned in previous articles about velocity and position.

To quickly review that…

To find how far an object moves—that is to say, to find the magnitude of an object's displacement—the rate of position change is multiplied by the amount of time that it changes. Or, as an equation,  

df = vt  

where d is change in position (displacement), v is rate of change in position, (average velocity), and t is elapsed time. 

In some cases, displacement is not measured from the point at which motion begins, but is measured from some other fixed point. In those cases, the total displacement becomes the sum of the initial displacement (initial position) and the distance moved. In such a case, 

df = di + vt 

where df is the final displacement from the fixed point, di is the initial displacement from the fixed point, v is rate of change in position, (average velocity), and t is elapsed time. (And where elapsed time the difference between final time and initial time.)

Finding Final Velocity

What is true for displacement closely parallels what is true for acceleration.

The change in velocity can be found by multiplying the rate that velocity changes (acceleration rate) by the amount of time that it changes (elapsed time). As an equation,

v = at

where

v is the change in velocity (sometimes written as ∆v),
a is the rate of acceleration, and
t is the elapsed time.

In many cases, an object is already in motion (has an initial velocity) when the acceleration begins. Logic and reason lead to the conclusion that the final velocity of an object is the initial velocity plus or minus the change in velocity. Since change in velocity is found (see above) easily, the equation for final velocity is

vf = vi + at

where vf is final velocity, vi is initial velocity, a is the rate of acceleration, and t is the elapsed time. (And where elapsed time the difference between final time and initial time.)

In many cases, however, Vi is 0 (zero).

Finding Average Velocity

To extend and begin to combine the above principles of motion, it is necessary to first consider how to find average velocity when given the rate of acceleration and time. Fortunately, this is relatively logical and easy to reason out. And the math on it is equally easy!

To apply logic, if something accelerates at a constant rate, it starts our at one velocity and moves faster and faster until it reaches a final velocity. Suppose it starts out a 0 m/s and ends up at 4 m/s. Through the course of its acceleration, it passes through all of the velocities between 0 and 4 m/s.

That might look like this:

0 sec, v = 0

2 sec, v = 1

4 sec, v = 2

6 sec, v = 3

8 sec, v = 4

The data above shows and elapsed time of 8 seconds, and in that time, the velocity of the object changed from 0 to 4 m/s. It might be tempting to just guess that, since the final velocity was 4 m/s, over the 8 seconds, the average velocity was 2 m/s. If that guess was made, the answer would be correct.

The list below gives Vf, Vi, and Vave:

 Vf= 0, Vi=6, Vave= 3

 Vf= 10, Vi=16,, Vave= 13

 Vf= 0, Vi=10, Vave= 5

 Vf= 100, Vi=160, Vave= 130

 Vf= 4, Vi=8, Vave= 6

Simply allowing logic and reason to stand alone would violate the need of science to describe everything with equations! So, there is an equation:

vave = (vf  + vi) / 2

So what if vi is zero? The math gets even easier! 

vave = (vf  + 0) / 2

vave = vf / 2

NOTES AND HINTS:

See also THIS.

The Equations:



df = (di + ∆d)

Use this if:

• You need to find the final distance, total distance, or final position AND You know BOTH di and how much d changes (e.g. it moves 10 meters)

vf = vi + ∆v

Use this if:

• You need to find the final velocity AND You know BOTH vi and how much v changes (e.g. speed increases by 10 m/s)

v(ave) = (vi + vf)/2

Use this if:

• You need to find the average velocity 
• You know BOTH vi and vf (you might have to calculate vf!)

vf = vi + (a)(t)    This will answer almost all “how fast” questions.

Use this if: 

• 2 velocities are given 
• You don’t have distance

df = di + (vi)(t) + (½)(a)(t²)   This will answer almost all “how far” questions.

Use this if:

• Only 1 velocity is given 
• You have a distance given (df is rarely 0, but sometimes it is the unknown)

The hints and helps:


It is vi if… 

“…traveling at a rate of…” “…moving at…”"…has a velocity of…"

If it is “at rest” then vi = 0 and di is probably 0

The words "begin" and "starts" generally go with the initial values.

If it is “moving at a constant” rate or if it “has a constant velocity” then a = 0


It is possible that something not given (but which is not the thing to be found) should have a value of 0 (zero).

Moving on to Displacement

The ability to find an average velocity allows more work with acceleration to be done. Since, if the average velocity is known, the displacement can be found. The work can be done in three steps. 

• First, find the final velocity.
• Next, find the average velocity.
• Last, use the average velocity and the elapsed time to find the displacement.

However, doing the work in three steps is not necessary, because math.

The  best, most direct, preferred process for the calculation of final distance/displacement will be revealed HERE.

Yet, just for the sake of having completed the thoughts above, an example is provided. 

EXAMPLE:

A car with an initial velocity of 20 meters per second (m/s) accelerates at a uniform rate of 3 (m/s)/s for 10 seconds. How far does it travel while it is accelerating?

Step 1.) Find the final velocity.

vf = vi + at

vf = 20 + 3 • 10

vf = 20 + 30

vf = 50 mph

Step 2.) Find the average velocity.

v(ave) = (vf + vi)/2

v(ave = (50 + 20)/2

v(ave) = 70/2

v(ave) = 35 mph

Step 3.) Find the displacement.  NOTE: the elapsed time is the same for displacement and acceleration.

df = di + vt

df = 0 + 35 • 10

df = 350 meters (350 m)

By combining an understanding of average velocity with the ability to find displacement and to find final velocity, it becomes possible to calculate the displacement that occurs during a period of acceleration.  This 3-step method can be used when given acceleration and time and asked to find distance.

Concepts of Acceleration

Introduction to Chemistry and Physics Index

Physics Index

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 

(df  - di) = v ( tf - 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

(vf  - vi) = a ( tf - 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!