Suppose a nail is stuck in the tire of a bike wheel with a diameter of 65 of cm. Let H(t) be the height of the nail, in cm, after t seconds of travel.

Question

Suppose a nail is stuck in the tire of a bike wheel with a diameter of 65 cm. Let H(t) be the height of the nail, in cm, after t seconds of travel.

Determine the equation of the axis and explain what this represents in the context of the problem.

Please organize your answer nicely and have it neatly. Thank you :D

Answer 1

I think the best way to model this scenario would be with the sine or cosine function because the height of the nail will continue to oscillate (go up and down) as the bike tire rolls. Let's go with the sine function, and assume:

H(t) = A*sin(B*t + C) + D

We know that:

max(H(t)) = 65

min(H(t)) = 0

max(sin(B*t + C)) = 1

min(sin(B*t + C)) = -1

So we can determine that H(t) = 32.5*sin(B*t + C) + 32.5, as this would result in a max and min of H(t) being 65 and 0.

For example, if sin(B*t + C) = 1, then H(t) = 32.5*(1) + 32.5 = 65

and if sin(B*t + C) = -1, then H(t) = 32.5*(-1) + 32.5 = 0

If we assume that the tire struck the nail on the ground when t = 0, then H(0) = 0

H(0) = 32.5*sin(0 + C) + 32.5 = 0

sin(C) = -32.5/32.5 = -1

C = 3pi/2 (assuming we are measuring in radians)

H(t) = 32.5*sin(B*t + 3pi/2) + 32.5

The unsolved variable B can be thought of as representing the velocity of the bike tire. For example, if B = 1, then it would take 2pi seconds (about 6.28 seconds) for the tire to make a full rotation because:

H(0) = 32.5*sin(3pi/2) + 32.5 = -32.5 + 32.5 = 0

H(2pi) = 32.5*sin(2pi + 3pi/2) + 32.5 = -32.5 + 32.5 = 0

If B > 1, then the tire is moving faster. If B < 1, then the tire is moving slower.

If the speed of the tire is not constant, then B would not be a constant. Perhaps the bike is decelerating because there is a nail in it, so we assume B is decreasing over time, so B = e^(-xt) (or something like that, where x is another variable that we are introducing). You can make this as complex as you like, I just picked an exponential function for example.

I hope this is helpful!