A particle moving along a straight line has velocity given for by v(t)=A B , (Bt)

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A particle moving along a straight line has velocity given for

t\ge 0

v(t)=A B\,\sin(Bt),

where

A

has units of length and

B

has units of reciprocal time; both are positive constants. (a) Derive a general expression for the position function

s(t)

of the particle by antidifferentiating

v(t)

. Include an arbitrary constant of integration

C

. (b) The particle is observed to be at the origin when

t=0

. Use this initial condition to determine the particular position function

s(t)

. (c) Find the smallest positive time

t>0

at which the particle returns to the origin. Show all reasoning leading to your answer and express the result in terms of

B