A weight attached to a spring is pulled down so that it is 10 cm from the floor and is released so that it bounces up and down. When the effects of friction and gravity are ignored. It's height can be modeled by a sine function of the time since it started bouncing. The weight reaches its first maximum height of 50 cm at 1.5 s.

Question

a. Write an equation for the height, in cm,of the weight as a function of time, in seconds.
b. graph the equation from part a.
c. When is the weight moving up fastest?
d. When is it moving down faster?
e. at what time is the weight changing direction.

Answer 1

the amplitude is (max-min)/2 = (50-10)/2 = 20

so, the center of the wave is at y=30, and it varies between 30+20 and 30-20

since it starts at a minimum at t=0,

h(t) = 30 - 20cos(kt)
since its half-period is 1.5 sec, k = pi/1.5 = 2pi/3
h(t) = 30 - 20cos(2pi/3 t)

since we want a sine function, and cos(v) = sin(pi/2 - v)

h(t) = 30 - 20sin(pi/2 - 2pi/3 t)
h' = 20cos(pi/2 - 2pi/3 t)*(-2pi/3)
h'' = -20sin(pi/2 - 2pi/3 t)(-2pi/3)(2pi/3)

moving fastest/slowest when h'' = 0
changes direction when speed=0, h' = 0