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.

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.

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

a. The height of the weight can be modeled by a sine function with an amplitude of 40 cm, since it reaches a maximum height of 50 cm and starts at a height of 10 cm. The equation for the height, h, as a function of time, t, in seconds is:

h(t) = 40sin(bt) + 10,

where b is a constant that determines the period of the sine function.

b. To graph the equation, you can plot the height as a function of time on a coordinate plane. The x-axis represents time (t) in seconds, and the y-axis represents the height (h) in centimeters. Use the equation h(t) = 40sin(bt) + 10 and the given information to plot the graph.

c. The weight is moving up fastest when its derivative with respect to time (velocity) is at a maximum. Taking the derivative of the equation h(t) = 40sin(bt) + 10 with respect to t, we get:

v(t) = 40bcos(bt).

The cosine function has a maximum value of 1, so the weight is moving up fastest when the cosine function is equal to 1. This occurs when bt = 0, so t = 0. Therefore, the weight is moving up fastest at t = 0 seconds.

d. The weight is moving down fastest when its velocity is at a minimum. The cosine function has a minimum value of -1, so the weight is moving down fastest when the cosine function is equal to -1. This occurs when bt = pi, so t = pi/b. Therefore, the weight is moving down fastest at t = pi/b seconds.

e. The weight changes direction when its velocity changes sign. In other words, when the velocity crosses the x-axis. The velocity is given by v(t) = 40bcos(bt), so it changes sign when cos(bt) = 0. This occurs when bt = pi/2, pi/2 + pi, 3pi/2, 3pi/2 + pi, etc. In general, the weight changes direction at times t = (2n + 1)(pi/2)/b, where n is an integer.