What does dv/dt mean, how is it used, and when?

[badr]

TL;DR

Derivative usage in basic functions.

Hi everybody.


I am Looking for brief explanation and some practical examples to practice the usage of the term dv/dt in modern applications .

 

[berkeman]

Welcome to PF.

What is "v" in this context? Voltage, velocity, or just any variable? What is your background in differential calculus so far? Where are you in school (what year and type of school)?

 

[Lnewqban]

Welcome, @badr !

One example could be the instantaneous linear velocity (respect to a very short time dt) of the piston in an internal combustion engine.

For a greater period of time (Δt), the value of that velocity will change from zero to a maximum value in a cyclic way.

 

[badr]

Welcome to PF.

What is "v" in this context? Voltage, velocity, or just any variable? What is your background in differential calculus so far? Where are you in school (what year and type of school)?

Thanks for the quick reply.

I have studied éléctro-téchnique , European term for electrical engineering.

I took courses in sine wave function expression.

I have good understanding of measurement but not analysis !

I didn't go much further in maths unfortunately , specially when it comes to functions and major calculus.

 

[Mark44]

I am Looking for brief explanation and some practical examples to practice the usage of the term dv/dt in modern applications .

I have good understanding of measurement but not analysis !
I didn't go much further in maths unfortunately , specially when it comes to functions and major calculus.

A brief explanation, but not one that's very helpful, is that dv/dt is the instantaneous rate of change of v with respect to t. The 'v' is usually velocity, and 't' is usually time. Since your background is more on the electronics side, the instantaneous rate of change of voltage with respect to time would normally use V, as in dV/dt.

The usual course in understanding derivatives (dv/dt and dV/dt are called derivatives) would be to study calculus. The first course in calculus, either a semester or quarter long, is where derivatives are defined and where examples of how they are used are presented. A second course usually presents integrals.

Other than attending a college or university, you might look into khanacademy.org, and specifically https://www.khanacademy.org/math/differential-calculus.

 

[jack action]

Let's imagine you are walking and at each hour you note the distance you have traveled:

10 h 00	0 km
11 h 00	1 km
12 h 00	4 km
13 h 00	9 km
14 h 00	16 km
15 h 00	25 km

What was your average speed for the entire run?
$$\frac{25\ km - 0\ km}{15\ h - 10\ h} = 5\ km/h$$
What was your average speed at each hour?

At 11 h 00:
$$\frac{4\ km - 0\ km}{12\ h - 10\ h} = 2\ km/h$$
At 12 h 00:
$$\frac{9\ km - 1\ km}{13\ h - 11\ h} = 4\ km/h$$
At 13 h 00:
$$\frac{16\ km - 4\ km}{14\ h - 12\ h} = 6\ km/h$$
At 14 h 00:
$$\frac{25\ km - 9\ km}{15\ h - 13\ h} = 8\ km/h$$
As you can see, the equation has the form
$$v = \frac{\Delta position}{\Delta time}\ or\ v = \frac{\Delta x}{\Delta t}$$
If we had taken measurements more often, we could have been even more precise as to what was your speed at any point during your walk. Imagine taking measurements every second. Then the $\Delta t$ would have been 2 seconds instead of 2 hours giving you practically the instantaneous velocity at each point of your walk.

So the definition of the instantaneous velocity $\frac{dx}{dt}$ would be the one where $dt$ tends to $0$ or:
$$\frac{dx}{dt} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$$
The $d$ stands for "small difference" as opposed to $\Delta$ (greek letter for $d$) which stands for "large difference".

In the following image, the velocities we calculated are represented by the slope of the yellow line, and the instantaneous velocity at $4:00$ is represented by the slope of the red line (which is tangent to the curve $f$).

 

[DaveE]

A common usage for EEs is in describing how capacitors work.
The current $i(t)=C\frac{dv}{dt}$, where $v(t)$ is the voltage and $C$ is the capacitance.

 

[badr]

View attachment 344171​

Thanks for this Nice graph. And your first equation gives more clarity.

Actually when i was in high school , the difficulty I always had is focusing my attention on the 2 axes of the graph ( x,y) or more (x,y,z) and working through the details of the curves , and all the discipline that comes out.

I am 34 years old now btw.

I am treating psychosis since teenage (18) , which makes it slightly challenging to hold focus in the long run.

Can you suggest some questions , and while i anwer them , i can move into small exercises in khan ?

 

[jack action]

Can you suggest some questions , and while i anwer them , i can move into small exercises in khan ?

I found this web page that explains the basics of derivatives along with some questions at the end.

 

[badr]

I found this web page that explains the basics of derivatives along with some questions at the end.

I appreciate your help jack. This will come in handy for sure.