The position of an object attached to a spring is given by s(t)=(1)/(4)cos(4t)-(1)/(6)sin(4t), where t is time in seconds. When 0, how many times is the velocity of the object equal to 0?

Question

The position of an object attached to a spring is given by s(t)=(1)/(4)cos(4t)-(1)/(6)sin(4t), where t is time in seconds. When 0, how many times is the velocity of the object equal to 0?

Zero

One

Two

Three

Four

Answer 1

s(t) = (1/4)cos(4t) - (1/6)sin(4t)

Velocity = 0 when s'(t) = 0. So, -sin(4t) - (2/3)cos(4t) = 0

(2/3)cos(4t) = - sin(4t)

tan(4t) = -2/3

Tangent is negative in quadrants 2 and 4.

4t = 2.55 + kπ (values are in radians)

So, t = 0.6375 + kπ/4, k = 0, ±1, ±2, ...

Plugging in k = 0, 1, 2, and 3, we get t = 0.6375, 1.4229, 2.2083, 2.9937

There are 4 solutions in the interval [0, π]