Velocity in Classical Mechanics by Douglas Gregory

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Discussion Overview

The discussion centers on the concepts of velocity and displacement as presented in Douglas Gregory's textbook on classical mechanics. Participants explore the definitions, relationships, and implications of these concepts, including their mathematical representations and units of measurement.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the reasoning behind the expression for velocity as the product of speed and the unit tangent vector, seeking clarification on the relationship between displacement and arc-length.
  • Another participant clarifies that arc-length is the length of a curve measured along the curve and provides an example related to circular motion.
  • There is a discussion on the importance of displacement in vector concepts, with some participants suggesting that vector concepts are easier to work with due to their abstract nature.
  • Participants discuss the measurement units for velocity and speed, noting that both can be expressed in different units such as meters per second or kilometers per hour.
  • Questions arise about the relationship between velocity and speed, specifically whether finding velocity is necessary to determine speed and vice versa.
  • One participant asserts that the absolute value of velocity corresponds to speed, while another suggests using "magnitude" instead of "absolute value" when referring to vectors.

Areas of Agreement / Disagreement

Participants express differing views on the terminology used for vectors and the relationship between velocity and speed. While some points are clarified, no consensus is reached on the broader implications of these concepts.

Contextual Notes

Participants express uncertainty regarding the definitions and relationships between displacement, velocity, and speed, as well as the appropriate terminology for vector magnitudes.

manimaran1605
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i am using textbook in classical mechanics by douglas gregory. It is known that rate of change of displacement with time is velocity , but it is given that the velocity at any instant is the equal to the product of speed and its unit tangent vector at that instant in vector form i.e, V=vT (this expression is reasonable), but the method he proves this expression questions me, he says that rate of change of displacement with time is velocity i.e, dR/dt=V then he said distance is changing by time and by chain rule (dR/ds)(ds/dt) was that mean 's' is also the function of displacement? if yes how?

another doubts: why it is displacement so important( i know vector concepts are abstract concepts)?, why the velocity is measured in meter per second and speed in kilometer per hour? we can use the distance as the measure of time but we can't use the displacement as the measure of time why?
 
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hi manimaran1605! :smile:
manimaran1605 said:
… he says that rate of change of displacement with time is velocity i.e, dR/dt=V then he said distance is changing by time and by chain rule (dR/ds)(ds/dt) was that mean 's' is also the function of displacement? if yes how?

no, s is the arc-length (see eg http://en.wikipedia.org/wiki/Arc_length)

arc-length is the length of a curve between two points of the curve, meaured along the curve

eg the arc-length of an arc of angle θ on a circle of radius r is rθ :wink:
another doubts: why it is displacement so important( i know vector concepts are abstract concepts)?,

because, vector concepts are abstract concepts that are very easy to work with
why the velocity is measured in meter per second and speed in kilometer per hour?

both velocity and speed can be measured in both units, ie metres per second or kilometres per hour
 
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can u give me a example how these vector concepts are easy to work? does that mean we are finding velocity to find speed (am i right)?
Another question: does the absolute value of the velocity at any instant give the speed at that instant?
 
manimaran1605 said:
does that mean we are finding velocity to find speed (am i right)?

you can do it in either direction … finding velocity to find speed, or finding speed to find velocity
does the absolute value of the velocity at any instant give the speed at that instant?

yes, the scalar, speed, v, is the absolute value of the vector, velocity, v :smile:
 
I would not use the term "absolute value" for a vector: "length" or "magnitude" would be better.
 

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