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3.1 Velocity Motion

Consider a rectilinear motion such as piston in the cylinder of an engine.
(rectilinear motion is motion along a straight line)

We represent a body in the form of a moving point M.

The distance s of the moving point from an initial position M₀ will depend on time t.

Hence, s = f(t)

At an instant of time t let the moving point M be a distance d from the initial position M₀.

At a later instance t + Δt let the point be at M₁ which is a distance s + Δs from the initial position of M.

Hence, over the interval Δt the distance s changed by Δs, we call these increments of time and distance respectively.

                              s                                 Δs
         ~~~~~~~~~~~~~~~~~~~~~~  ~~~~~~~~~~~
       M₀                                          M                  M₁
------0-----------------------------------0----------------0------
                                                       --------------->
                                                                v

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Average Velocity : :
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Consider the ratio Δs/Δt, that is the ratio of of an increment of distance given an increment of time.

It gives us an average velocity of motion of the point M during time Δt, vₐᵥ = Δs/Δt.

Why do we say that the average velocity is equivalent to this ratio Δs/Δt?

As a purely mathematical consideration, Δs/Δt really is an average in the statistical sense. Δt is a sample size, that is a representation of the count of total occurrences. Δs is the count of a specific set of occurrences. There isn't a discernible difference between the mathematical object x̄ = ∑xᵢ/N and Δs/Δt given Δs = ∑s.

Physically, Δt is the count of all occurrences represented as a moment in time. Δs represents the count of a specific set of occurrences which are measurable changes of distance. Each measurable change of distance can be mapped ont-to-one with a change in time, hence mathematically the set of occurrences that are a increment of distance is a subset of the total number of possible occurrences.

Velocity is how much distance a point travels over a period of time, hence we are concerned with the physical quantities s and t. It is easy to measure the distance something has moved over a period of time, hence the ratio Δs/Δt is a natural formula to come up with to describe this idea of motion over a period of time. However, it does tell you about velocity at any given point of time, just the velocity over an extended period.

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Instantaneous Velocity : :
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The quantity Δs/Δt does not provide details about how the increment the motion varied over Δt, that is if the particle covered more distance at the beginning of Δt and less distance at the end or whatever else can be imagined.

To derive information like this directly from the ratio Δs/Δt we have compute the ratio for those smaller increments of time Δt, that is we have to take smaller and smaller increments of Δt.

This procedure in which we take smaller and smaller increments is known as a limiting process, mathematically to achieve a limiting process we of course use the limit.

By the limit we compute the vₐᵥ as Δt approaches 0,
vᵢₙₜ = lim (Δt -> 0) Δs/Δt

Hence the velocity of motion at a given instant is the limit of the ratio of the path Δs to increment of time Δt as the time increment approaches zero.

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Increment as Function : :
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Increments just changes in a quantity.

Δs is a change in distance from one position M to another position M₁ and we describe the path that is distance via the function s = f(t).

Hence, any point on a distance path in a coordinate grid can be represented via a coordinate (t, s).

If the point M is represented by (t,s), then the point M₁ can be represented as (t+Δt,s+Δs) given we know the difference between M and M₁ is an increment.

The function s = f(t) corresponds to point M and therefore the coordinate (t,s), while the point M₁ corresponding to the coordinate (s+Δs,t+Δt) is represented by the function s+Δs = f(t+Δt).

To measure the distance between M and M₁ which is what Δs physically corresponds to we simply take the difference between the functions s=f(t) and s+Δs = f(t+Δt).

(s + Δs) - s = f(t + Δt) - f(t) -> Δs = f(t + Δt) - f(t)

As such, the functional form of the instantaneous velocity is, lim(Δt->0)[f(t+Δt)-f(t)]/Δt

Ex. Find the velocity of uniformly accelerated motion at an arbitrary time t and at t = 2 sec if the relation of the path traversed to the time is expressed as the formula s = (1/2)gt².
Solution.
At time t we have s = 1/2gt² while at time t+Δt we have s+Δs=(1/2)g(t+Δt)² = (1/2)g(t² + 2tΔt + Δt²)
Hence, Δs =(1/2)g(t² + 2tΔt + Δt²) - (1/2)gt² = gtΔt + (1/2)gΔt²
We form the ratio, [Δs/Δt = gtΔt + (1/2)gΔt²]/Δt = gt + (1/2)gΔt
We take the limt, vᵢₙₜ = lim(Δt->0)Δs/Δt = lim(Δt->0)[gt + (1/2)gΔt] = gt
Hence, vᵢₙₜ=gt and at t=2 sec where g = 9.8 m/sec² we have vᵢₙₜ = (9.8)92) = 19.6 m/sec

3.2 Definition of Derivative

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Definition : :
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Let there be a function y=f(x) defined in a certain interval.

Let the argument x receive an increment Δx.

Then the function y will receive an increment Δy.

Hence, y + Δy = f(x + Δx)

Let us find the increment of the function Δy,
Δy = f(x + Δx) - y = f(x + Δx) - f(x)

Form the ratio of the increments,
Δy   f(x + Δx) - f(x)
-- = ------------------
Δx           Δx

We find the limit of the ratio as Δx->0. If this limit exists it is called the derivative of a function f(x) denoted f'(x)
                            Δy                      f(x + Δx) - f(x)
f'(x) = lim(Δx->0) ---- = lim(Δx->0) ------------------
                            Δx                              Δx

The derivative is also denoted as y', yₓ', dy/dx

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Examples : :
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Ex. Given the function y=x²; find the derivative y'

1) at an arbitrary point x,
Suppose y=x²
let the argument x receive an increment Δx
For the increment Δx there is a corresponding increment Δy
It follows, y + Δy = (x + Δx)²
Let us find the increment of the function Δy,
Δy = (x + Δx)² - y = (x + Δx)² - x²
     = (x² + 2xΔx + Δx²) - x²
     = 2xΔx + Δx²
     = Δx(2x + Δx)
Form the ratio of the increments,
Δy      Δx(2x + Δx)
---- = ---------------- = 2x + Δx
Δx             Δx
Find the limit of the ratio as Δx->0,
                  Δy
lim(Δx->0) ---- = lim(Δx->0) (2x + Δx) = 2x + 0 = 2x
                  Δx

2) at x = 3
When x = 3 we have y'|ₓ₌₃ = 2(3) = 6

Ex. y = 1/x; find y;
Suppose y = 1/x
let the argument x receive an increment Δx
corresponding to the increment Δx is the increment Δy
it follows, y + Δy = 1/(x + Δx)
let us find the increment of the function Δy
Δy = 1/(x + Δx) - 1/x = (x - x - Δx)/(x(x + Δx)) = -Δx/(x(x + Δx))
Form the ratio of the increments,
             - Δx
           ----------
 Δy     x(x + Δx)            - Δx                  -1
---- = -------------- = ----------------  =  ------------
Δx            Δx           Δx(x(x + Δx))       x(x + Δx)
Find the limit of the ratio as Δx->0,
                  Δy                         -1               -1           -1
lim(Δx->0) ---- = lim(Δx->0) ----------- = ---------- = -----
                  ΔX                      x(x + Δx)     x(x + 0)      x²

3.3 Geometric Meaning of the Derivative

We now give a geometric interpretation of the derivative.

We must define a line tangent to a curve at a given point.

We take a curve with a fixed point M0 on it.

Taking a point M1 on the curve we draw the secant line M1M0.

We can take a point M1' on the curve and draw the secant line M1'M0

Again, we can take a point M1'' on the curve and draw the secant line M1''M0

If in the limitless approach of the point M1 along the curve to the point M0, the secant tends to occupy the position of the definite straight line TM0 which is called the tangent line.

Let us consider the function f(x) = x² and its corresponding curve in a rectangular coordinate system.

We see that the point M1 approaches the point M0 with each iteration of the secant line,

If we let M₀ be the point M₀(x,y) we can showcase this approach of M1 to M0 via the process of incrementing x by Δx.

M1 is the always the point M₁(x+Δx, y+Δy), that is M1 is always an some increments Δx and Δy away from M0

As M1 approaches M0, the increments Δx and Δy will decreases and once the secant line becomes the tangent the increments will be infinitesimals.

It can be see that there is a relationship between the angle α and the magnitudes of the increments Δx and Δy, that is the ratio Δy/Δx is described by the tangent of the angle α.
                                                         Δy
tanα = lim(Δx->0)tanφ = lim(Δx->0) ----- = f'(x)
                                                         Δx

Hence, f'(x) = tanα

ex. Find the tangents of the angles of inclination of the line tangent to the curve y=x² at the point M₁(1/2,1/4);
M₂(-1,1).
tanα₁ = y'|ₓ₌₀.₅ = 1 and tanα₂ = y'|ₓ₌-₁ = -2

Have you done much kinematics?

The displacement - velocity - acceleration relationship blew my mind when we first learnt it. Calculating the deceleration of a rocket at point x is the shit. 

Outro said: 

Have you done much kinematics?

The displacement - velocity - acceleration relationship blew my mind when we first learnt it. Calculating the deceleration of a rocket at point x is the shit. 

Yes, once upon a time I went through the first volume of Resnick/Halliday but that was years ago.

I am currently reading Physics for Mathematicians by Spivak and Theoretical Mechanics by Targ, both of which have their own corresponding threads.

I touch upon the the relationship between displacement and velocity in this thread in my first post under the section 'Average Velocity'. Upshot, it is interesting that limiting processes of ratios, both mathematically and physically, are just averaging processes. I believe this makes more sense to anyone with a computer science background given numerically this is how the approximation of floating point representations of the same mathematical objects works.

Adelaide said: 

Outro said: 

Have you done much kinematics?

The displacement - velocity - acceleration relationship blew my mind when we first learnt it. Calculating the deceleration of a rocket at point x is the shit. 

Yes, once upon a time I went through the first volume of Resnick/Halliday but that was years ago.

I am currently reading Physics for Mathematicians by Spivak and Theoretical Mechanics by Targ, both of which have their own corresponding threads.

I touch upon the the relationship between displacement and velocity in this thread in my first post under the section 'Average Velocity'. Upshot, it is interesting that limiting processes of ratios, both mathematically and physically, are just averaging processes. I believe this makes more sense to anyone with a computer science background given numerically this is how the approximation of floating point representations of the same mathematical objects works.

Nice, is your background compsci? 

I don't remember much of the limiting notation because we mainly rote learnt the derivation rules and just went from there, I learnt through an engineering lens so we're less concerned with the process and more with the outcome. 

1.4 Differentiability of Functions

Definition (Differentiable)
If the function function y=f(x) has a derivative at the point x=x₀, that is, if there exists
                                         Δy                         f(x₀ + Δx) - f(x₀)
                       lim(Δx->0) ----- = lim(Δx->0) ----------------------
                                          Δx                                 Δx

we say that for the given value x=x₀ the function is differentiable or has a derivative.

Theorem (continuity)
If a function y=f(x) is differentiable at some point x = x₀, it in continuous at this point.

So, functions cannot have derivatives at points of discontinuity.

Ex. A function f(x) is defined in an interval [0,2] as follows,
f(x) = x        when 0 <= x <= 1
f(x) = 2x - 1 when 1 < x <= 2
At x = 1 this function has no derivative despite it being continuous at this point.
When Δx < 0 we get,
                  f(1 + Δx) - f(1)                      [2(1 + Δx) - 1] - [2(1) - 1]                        2ΔX
lim(Δx->0) ------------------ = lim(Δx->0) -------------------------------- = lim(Δx->0) ------ = 2
                           Δx                                                  Δx                                        ΔX
When Δx > 0 we get,
                  f(1 + Δx) - f(1)                         [1 + Δx] - 1                        Δx
lim(Δx->0) ------------------- = lim(Δx->0) ----------------- = lim(Δx->0) ----- = 1
                           Δx                                         Δx                               Δx
We see that the limit depends on the sign of Δx and as such the function has no derivative at the point x = 1.
Geometrically, this is in accord with the fact that at the point x=1 the given point does not have a definite line tangent to it.

Outro said: 

Adelaide said: 

Outro said: 

Have you done much kinematics?

The displacement - velocity - acceleration relationship blew my mind when we first learnt it. Calculating the deceleration of a rocket at point x is the shit. 

Yes, once upon a time I went through the first volume of Resnick/Halliday but that was years ago.

I am currently reading Physics for Mathematicians by Spivak and Theoretical Mechanics by Targ, both of which have their own corresponding threads.

I touch upon the the relationship between displacement and velocity in this thread in my first post under the section 'Average Velocity'. Upshot, it is interesting that limiting processes of ratios, both mathematically and physically, are just averaging processes. I believe this makes more sense to anyone with a computer science background given numerically this is how the approximation of floating point representations of the same mathematical objects works.

Nice, is your background compsci? 

No, I pulled that little insight from a book called Numerical Methods using Matlab by Mathews and Fink. 

I am a math major.

I don't remember much of the limiting notation because we mainly rote learnt the derivation rules and just went from there, I learnt through an engineering lens so we're less concerned with the process and more with the outcome. 

Engineering is cool, for whatever reason I find thin film engineering interesting.

I really don't know why.

What kind of engineering?

I assume most calculus classes follow from a similar perspective, they after all need to keep everyone from engineering to mathematics students engaged with work relevant to their respective majors. I am reading Piskunov for that reason, it gets more into the nuts and bolts via proofs and is a great way to transition to Analysis which I will be taking next year.