   Chapter 3.4, Problem 87E

Chapter
Section
Textbook Problem

A particle moves along a straight line with displacement s(t), velocity v(t), and acceleration a(t). Show that a ( t ) = v ( t ) d v d s Explain the difference between the meanings of the derivatives dv/dt and dv/ds.

To determine

To show: The acceleration of the particle a(t)=v(t)dvds and to explain the difference between the derivatives dvdt and dvds.

Explanation

Given:

Displacement of the particle is s(t), velocity of the particle is v(t) and acceleration of the particle is a(t).

Proof:

Note that,

The velocity of the particle is v(t)=dsdt (1)

The acceleration of the particle is a(t)=dvdt (2)

From equation (2),

a(t)=dvdt=dvdsdsdt   (by chain rule

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