Height as a function of time word problem (help)!

Question

A reflector is fastened to the front wheel of a bicycle 20 cm from the center oF the wheel. The diameter of the wheel and inflated tire is 70 cm. If the bike is traveling
at 10 km/h, express the height of the reflector above the ground as a fuction of time. Assume that at
time t = 0 seconds, the reflector is at its highest point.

Answer 1

Bianca,

This is exactly the same curve we just describe for the sound wave except that is translated up 35 cm, the amplitude is 20 cm, the velocity is 10 kph. Because they have proscribed that the reflector starts at the highest point, the equation will be simpler to write using cosine instead of sine. We also need to get everything in the same units.

 

y = A*cos(bx) + c     y = ht above ground, A = amplitude, b = 2πƒ, x = time (s)

 

A = .2 m

v = (10kph)*(1000m/km)/(3600s/hr) = 2.8 m/s

To find the frequency, we need to look at how often the tire makes a complete rotation. The reflector will go around with the same frequency.

 

λ = circumference of the tire = πd = .7π

 

ƒ = v/λ = (2.8m/s)/(.7πm) = 4/π hz

 

 b = 2π(4/π) = 8

 

c = translation up from ground to the center of the wheel = .35 m

 

y = .2 cos(8x) + .35