Velocity in Classical Mechanics by Douglas Gregory

[manimaran1605]

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?

 

[tiny-tim]

hi manimaran1605!

… 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

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

 

[manimaran1605]

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?

 

[tiny-tim]

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

 

[HallsofIvy]

I would not use the term "absolute value" for a vector: "length" or "magnitude" would be better.