Calculus Problem Question, Elastic band

Question

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically.

The equation of motion is s(t) = 2cos(t) + 3sin (t) , t >= 0 , where s is measured in centimeters and t in seconds. (We take the positive direction to be downward.)

(a) Find the velocity function.

(b) When does the mass pass through the equilibrium position for the first time?

(c) How far from its equilibrium position does the mass travel?

Answer 1

(a) The velocity function is the derivative v(t) = s'(t) = -2sin(t) + 3cos(t).

(b) The mass passes through the equilibrium when the position s(t) = 0, so you need to solve 2cos(t) = -3sin(t) or tan(t) = -2/3. With a calculator, t = arctan(-2/3) = -0.59, but that doesn't make sense since t >= 0, so add π, the period of tangent: t = -0.59 + 3.14.

(c) The mass reaches its farthest point when the velocity is zero (since it changes direction there), so solve s'(t) = 0: tan(t) = 3/2. Using a calculator, t = arctan(3/2) = 0.98. Now find s(0.98) for the largest distance.