Displacement, Velocity, Acceleration.

In physics, we often want to know if things are moving and if so, how fast, in what direction, and how fast they are changing speed.

All these concepts can be summed up as Displacement, Speed, Velocity and Acceleration.

We will first start with Displacement.



DISPLACEMENT

Displacement is the amount of "space" that an object moves.

The starting position of any object that moves from that position is called its initial position and the final position that an object moves is called, well, you guessed it, its final position.

We can denote initial position with "i" and final position with "f."

So imagine we have a starting position denoted by Pℹ︎ and we have a final position denoted by Pƒ

If we subtract Pƒ from Pℹ︎ then we will get our displacement.

This can be written as  Displacement = Pƒ - Pℹ︎

or as: 

d = Pƒ - Pℹ︎ 


So you can imagine if we have some Number line where our starting position is at 1 and our final position is at 6. You can simply plug in the values and do the subtraction to get the displacement.

This object was displaced by 5.

If we wanted to now know how much time has passed since this object was first displaced, we can call that speed and if we wanted to know also what direction this object was moving in, then we call this Velocity.


VELOCITY / SPEED

The only difference between Speed and Velocity is that Speed is a Scalar value and Velocity is a Vector Value. You can read here about Scalar and Vector Values.

So if we wanted to know how much time has passed, we can imagine that we need a second number line that represents how much time has passed during this displacement.

 


We can do the same subtraction method that we did for displacement in space that we can now do for displacement of time.

time = tƒ - tℹ︎
or 

t = tƒ - tℹ︎

If we plug in the values, We will get:

t = 5 - 1 
t = 4


So the time it takes for our displacement in space is 4 units of time, which can be seconds, minutes, hours, etc. whatever the unit of measurement that is necessary.

Now, we have a ratio, which is distance traveled over time taken.

This ratio is distance over time or d/t
or ⍳(for length) / time (t). 

You might recognize this as Miles per hour if you are a driver in the United States, Or, Kilometers per hour if you live anywhere else.

That means if you are driving 35 miles an hour, then your initial position was 0 miles and your final position was 35 miles and your initial time was 0 hours and your final time was 1 hour, which can be written as:

Speed (s) =  35mi - 0mi / 1hr - 0hr  

When subtracting final values from initial values, we can just call these a "change in values" which can simply be represented by Greek Letter for Delta "△"

Now, we can re-write this as: S = △d / △t

This simply mean that speed is equal to a change in distance over a change in time. 

However, since "d" is often used for Derivatives, instead we will use the letter "⍳" for length/distance.

And, since in Physics, we are also interested in denoting not just magnitude but also direction, we should use velocity instead of speed.

Now, when we re-write the equation we get:

Velocity(v) = pƒ - pℹ︎  / tƒ - tℹ︎


Velocity equals the change in the position divided by the change in time

or

V = △⍳ / △t

IN THE REAL WORLD, VELOCITIES ARE NOT CONSTANT. THEY CAN BE, BUT MOST OFTEN VELOCITIES ARE CONSTANTLY CHANGING.

For example, as you are driving down the road, unless you are on Cruise Control, you are probably constantly changing speed by a few miles per hour every instant.

At one instant of time, you might be going 35 miles an hour, but then another instant of time you are going 34mph, then 38mph, then 30, then 40, maybe you see a cop and slow it all the way down to 25mph for a brief moment.

All of these changes in velocity are going to affect the final duration of your trip. By the time you are done with your journey, you may have finished with an average velocity of 33 miles per hour.

However, most drivers wouldn't think of it in this way, instead they would say I arrived at my destination early, late or on time.

So even though we may have changed how fast or slow we were driving over time throughout or trip, we may still want to know what our Average Velocity had been. 


AVERAGE VELOCITY

Finding your average velocity is like finding an average of anything else. You add up all of your values and you divide by the amount of values that you have.

If have values such as: 6, 7, 8

You would add them up & divide by 3: 

6+7+8= 21/3 = 7.

We have an average of 7.

However, when it comes to finding the Average Velocity, all we need to do is add our initial velocity to our final velocity and divide by 2 since that is our only 2 values. If you have more velocity values, you can divide by 3 or 4 or ever how many velocity values you recorded.


EXAMPLE: 

Imagine our driver had an initial velocity of 35 mph and had a final velocity of 20 mph.
We simply add these 2 velocities and divide by 2.

which would be: 55mph / 2  = 27.5 mph

So We have an AVG. VELOCITY OF 27.5 MPH

We can denote AVG. VELOCITY by writing: ⊽

* NOTE* - SPEED USES THE LOWER CASE v AND VELOCITY USES UPPER CASE V


So the Avg. Velocity can be written as:

V(avg.) = Vi + Vƒ / 2


or 

⊽ = Vi + Vƒ / 2

After the avg. Velocity is taken. Your equation would simply look like a regular velocity equation. We can see this by expanding out the terms 

Whereas after doing the math Avg. Velocity is still just distance over time  

⊽ = d / t 

As in our example where our Avg. Velocity was 27.5 mph.

It Seems we can now write avg. velocity in 2 different ways. 

Either the long Form:⊽ = Vi + Vƒ / 2

or the short form

⊽ = d / t  

So If AVG. VELOCITY CAN be equal to both the short form and the long form then both the short form and the long form must also be equal as long as we remember that the short form is not just a singular velocity but an avg. velocity.

We can write our equation like so:

d / t  =  Vi + Vƒ / 2

This is an important form to remember because this form will help us derive other forms needed when we ask our next question that we have previously brought up, which is what happens when velocities change throughout your journey. 

CHANGING VELOCITIES ARE KNOWN AS ACCELERATION. 



ACCELERATION 

As per our example, imagine we started off with a velocity of 20MPH, but then at some point during our Journey, we sped up to 40MPH.

What we have just done is ACCELERATE.  

That means we are going to get to where we were going at a faster rate.

Just like Velocity is the change of Distance over the change of time.

ACCELERATION is the change of VELOCITY over the change of TIME.

Let's take a look at how that's written.

VELOCITY:           V = △⍳ / △t

ACCELERATION:  a = △v / △t

A change in velocity means that we have now a final velocity minus an initial velocity.

But since "V" is equal to △⍳ / △t

We can just plug in change in distance over change in time into the acceleration equation.

a = △v / △t
V = △⍳ / △t

After you plug in your velocity into your acceleration, you will get this:

a = △⍳ / △t / △t

Now you have an equation with 2 changes in time, so you know of course that anytime in math, you want to reduce and combine like terms.

So after a little bit of Algebra, you will get:

a = △⍳ / △t ²

You can say acceleration is really the change in distance divided by the change time which are divided by the change in time again. 

Re-written, Acceleration is simply the change in distance divided by the change in time squared. 

or 

a = ⍳/t ²

or in it's expanded form

a = vƒ - v𝓲  / tƒ - t𝓲




FINAL VELOCITIES 

Sometimes in a physics problem, we might not know what the acceleration is, but other times we might now the acceleration, but not know what the final velocity is. Therefore, we need to re-arrange our equation such that we can solve for the final velocity.

Now, we need to take our equation and do some Algebra so that we can get something that is equal to the final velocity.

a = vƒ - v𝓲  / tƒ - t𝓲


In math, we know that numerators are really themselves over 1 multiplied by 1 over the denominator.


EXAMPLE:   5/2  is really   (5/1)(1/2)

Also we know that any number equal to a ratio (fraction) is also the denominator multiplied by that number to give you the numerator when re-written.

EXAMPLE:   2 = 6/3  or  2(3) = 6

We can Apply these rules to our acceleration equation. 

(tƒ - t𝓲 ) a = vƒ - v𝓲

Now, we just need to move our Initial Velocity to the other side of the equation so that our Final Velocity will be by itself.

Now we get an equation that looks like this. 

vƒ = a (tƒ - t𝓲 ) + v𝓲

Next, we distribute our acceleration and just move our initial velocity to the other side 

vƒ = v𝓲  + (a)tƒ - (a)t𝓲 

We subtract one acceleration times time from the other acceleration times time and we are left with only 1 acceleration times time which give us: |

The final Velocity is equal to the initial velocity plus acceleration times time.vƒ = v𝓲  + (a)t 


Now, we have 2 Derived Equations.Avg. 

Velocity
d/t = v𝓲  +  vƒ  / 2

Final Velocity

vƒ = v𝓲  + (a)t 



Because both derived equations have the final velocity term. We can now plug in the final velocity equation into the Avg. Velocity equation.

d/t = v𝓲  +  v𝓲  + (a)t  / 2

SOLVING FOR DISTANCE  


Now that we have plugged in the final velocity into the avg. Velocity, we can now create an equation for distance.

We just have to do some more Algebra.

(t/1) (d/t)  =  (v𝓲  + v𝓲  + (a)t / 2)(t/1)

Now, we cross cancel and multiply.

d = (2v𝓲 +  (at) ) (t) / 2


Now, after a couple more calculations, we finally have our distance equations. 

d = v𝓲 (t) +  1/2 at²

Now, we have 3 Derived Equations.


Avg. Velocity
d/t = v𝓲  +  vƒ  / 2

Final Velocity

vƒ = v𝓲  + (a)t 

Distance
d = v𝓲 (t) +  1/2 at²

The last equation that we need to find is the one for time.

There are 2 different ways that we can find a time equations.

First let's set avg. velocity equal to time. 

d/t = v𝓲  +  vƒ  / 2


The first thing we need to do is isolate time from distance. After some more algebra, we can do this by cross multiplying.

d/t = v𝓲  +  vƒ  / 2

d(2) = t(v𝓲  +  vƒ)

Now, we divide to isolate time

 t = d(2)/(v𝓲  +  vƒ) 


2nd let's set final velocity equal to time. 

vƒ = v𝓲  + (a)t

 -(a)t + vƒ = v𝓲 

 -(a)t  = v𝓲  - vƒ

t  = v𝓲  - vƒ / -(a)

t  = -v𝓲  + vƒ / a

Now, we can set the two time equations equal to each other


-v𝓲  + vƒ / a = 2d/(v𝓲  +  vƒ) 

We want to do this to get our last equation which is the final velocity squared equation.

After doing some more Algebra, you should get vƒ²on one side by itself by cross multiplying.


(-v𝓲  + vƒ)(v𝓲  +  vƒ)  =  2d(a)

Then we multiply our binomials


-v𝓲² - (v𝓲 )(vƒ) + (v𝓲 )(vƒ) + vƒ²  = 2d(a)
-v𝓲² + vƒ² = 2d(a)
vƒ² = 2d(a) + v𝓲²
vƒ² = v𝓲² + 2da

Now, we have 5 Derived Equations.

Avg. Velocity
d/t = v𝓲  +  vƒ  / 2

Final Velocity

vƒ = v𝓲  + (a)t 

Distance
d = v𝓲 (t) +  1/2 at²

Time

t  = -v𝓲  + vƒ / a

t = d(2)/(v𝓲  +  vƒ)

Final Velocity Squared

vƒ² = v𝓲² + 2da

With these 5 equations, we should be able to solve any velocity or acceleration problems that come our way. We could solve for Time, Avg. Velocity, Final Velocity, Distance or Final Velocity Squared.