   Chapter 7.2, Problem 65E

Chapter
Section
Textbook Problem

# A particle moves on a straight line with velocity function v ( t ) = sin ω t   cos 2 ω t . Find its position function s = f ( t ) if f ( 0 ) = 0.

To determine

To find: the position function given a velocity function of a particle moving in straight line.

Explanation

Recall that velocity is the rate at which the position of an object changes. If the position of a particle is known as a function of time, then its velocity will be the rate of change of its position. Therefore, velocity function is the derivative of position function.

Formula used:

Let the position function of a particle moving in straight line be s(t) and its velocity function be v(t). Then, by definition v(t)=ddt(s(t)). In other words, position function is the anti-derivative of velocity function. So, if velocity function is given, the position function will be:

s(t)=v(t)dt

Given:

Velocity of the particle at time t, v(t)=sinωtcos2ωt

Initial condition f(0)=0

Calculation:

Substitute the expression for velocity v in the formula for position

s=f(t)=v(t)dt=sinωtcos2ωtdt

Use the substitution u=cosωt,du=ωsinωtdt

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