What is the difference between Δv/Δt and dv/dt in calculus?

In summary, the difference between Δv/Δt and dv/dt is that Δv/Δt represents the average change in velocity over a non-zero time interval, while dv/dt represents the instantaneous rate of change in velocity at a specific moment in time. The limit concept in calculus allows us to determine this instantaneous rate of change by making the time interval smaller and smaller until it approaches 0.
  • #1
urbano
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I know delta means change , but I don't what the difference between Δv/Δt vs dv/dt is ?

I am at the noob end of calculus so trying to grasp how to interpret things like dv/dt or what what dv would mean if it were standing alone.

TIA
 
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  • #2
1. When we use "delta" rather than "d", we mean that, say "delta v"/"delta t" is the average change in "v" over the non-zero time interval "delta t".
2. Now, dv/dt is what we call the instantaneous rate of change in "v", that is the limit of "delta v"/"delta t" as we let the time interval "delta t" shrink to zero.
 
  • #3
The "delta t shrinking to 0" is the limit process:
[tex]\dfrac{dx}{dt}= \lim_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t}[/tex]

Calculus texts typically spend a good deal of time on the "limit" concept.
 
  • #4
Suppose that at time ##t_0## a car starts at a point ##x_0## and travels to point x arriving at time t. Then the average velocity of the car over its trip is ## \Delta x/ \Delta t = \frac {x-x_0}{t-t_0}## -- that is the change in distance divided by the change in time.

This tells you nothing about the velocity at any given moment. The car likely started at 0, gained speed up to a certain point, stopped for a light, got lost and had to go back a block, etc. So if you want to know how fast the car was traveling, and in what direction, at any given moment, ## \Delta x/ \Delta t ## tells you almost nothing.

But if you look at the average velocity of a small time period, that is closer to the velocity at any given time ##t_0## in the period. Make the time period smaller yet, and you are closer yet to the velocity at ##t_0##.

The genius of calculus was to see that you can let that time interval go to 0 and ## \Delta x/ \Delta t## may "converge" to a simple number -- say 30 mph. That is dx/dt -- the instantaneous velocity.
 

1. What is "Delta V" and "Delta t" in the context of physics?

"Delta V" refers to the change in velocity of an object, while "Delta t" refers to the change in time. They are both measures of change and are commonly used in physics to calculate acceleration and other motion-related quantities.

2. How is "Delta V/ delta t" different from "dv/dt"?

"Delta V/ delta t" is the average rate of change in velocity over a specific time interval, while "dv/dt" is the instantaneous rate of change in velocity at a specific moment in time. In other words, "Delta V/ delta t" is a measurement of overall change, while "dv/dt" is a measurement of change at a specific point.

3. Can "Delta V/ delta t" and "dv/dt" have different values for the same object?

Yes, they can have different values. "Delta V/ delta t" is calculated by taking the change in velocity over a specific time interval, while "dv/dt" is calculated at a specific moment in time. Depending on the object's acceleration, these values may differ.

4. How are "Delta V/ delta t" and "dv/dt" used in real-world applications?

These measurements are commonly used in fields such as physics, engineering, and aerospace to calculate the acceleration of objects, such as rockets and spacecraft. They are also important in understanding the motion of everyday objects, such as cars and trains.

5. What is the significance of "Delta V/ delta t" and "dv/dt" in the study of motion?

These measurements are essential in understanding the kinematics of an object, which is the study of an object's motion without considering the forces acting on it. They also play a crucial role in determining an object's acceleration, which is important in predicting and controlling its future motion.

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