A cyclist is travelling in a straight path at a constant speed along a dirt road with a button stuck to the outside of one of their wheels. The movement of the button (as viewed from the side) can be described using the equations of a cycloid, x=r(t−sint), y=r(1−cost) where t represents the angle through which the wheel has rotated and r represents the radius of the wheel. a) Every time the button touches the ground, it leaves a mark on the dirt. If the marks are 64π cm apart, what is the radius of the tire wheel? b) What value(s) of t correspond to the button touching the ground? c) Show that when t = 2π, we have dx/dt = 0. Does this mean that the cyclist is stopped? Why or why not?
A cyclist is travelling in a straight path at a constant speed along a dirt road with a button stuck to the outside of one of their wheels.
The movement of the button (as viewed from the side) can be described using the equations of a cycloid,
x=r(t−sint), y=r(1−cost)
where t represents the angle through which the wheel has rotated and r represents the radius of the wheel.
a) Every time the button touches the ground, it leaves a mark on the dirt. If the marks are 64π cm apart, what is the radius of the tire wheel?
b) What value(s) of t correspond to the button touching the ground?
c) Show that when t = 2π, we have dx/dt = 0. Does this mean that the cyclist is stopped? Why or why not?
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