A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos(2(pi)t). Determine the particle's maximum velocity.

Answer 1

max velocity occurs when the second derivative is zero

so ...
x(t) = 2πt + cos(2πt)
x '(t) = 2π - 2π(sin 2πt)
x ''(t) = 0 - 4π^2(cos 2πt)
= 0 for a max/min of velocity
cos 2πt = 0
2πt = π/2 or 2πt = 3π/2
t = 1/4 or t = 3/4

if t= 1/4
v(1/4) = x '(1/4) = 2π - 2π(sin π/2) = 2π - 2π = 0
if t=3/4
v(3/4) = 2π - 2π(sin 3π/2) = 2π + 2π = 4π