# Calculus 1/parametric equations question?

The motion of a point on a bicycle wheel is described by the parametric equa-

tions below. A and B are positive constants. Time is measured in seconds and distance in

meters (its a big wheel).

x(t) = At + cos(2t)

y(t) = B − sin(2t).

(c) (3 points) Use the fact that the horizontal velocity of the point is zero when it is at the

bottom of the wheel to find the constant A.

(d) (3 points) Find the speed of the point when it is at the top of the wheel. Hint: If you

couldn’t do parts (b) or (c) you may include the letters A and/or B in your answer .

Relevance
• Pope
Lv 7
2 months ago

In x(t) = At + cos(2t), 2t must be the clockwise rotation of the trace point from the 3 o'clock position. Let the point be on bottom.

2t = 2kπ + π/2, for some integer k

x(t) = At + cos(2t)

x'(t) = A - 2sin(2t)

When 2t = 2kπ + π/2, x'(t) = 0.

A - 2sin(2kπ + π/2) = 0

A - 2(1) = 0

A = 2

Again, 2t must be the clockwise rotation of the trace point from the 3 o'clock position. Let the point be on top.

2t = 2kπ - π/2, for some integer k

x'(t) = A - 2sin(2t)

x'(t) = 2 - 2sin(2t)

x'(2kπ - π/2) = 2 - 2sin(2kπ - π/2) = 2 - 2(-1) = 4

There is a geometrically intuitive way to reach this same answer. Clearly the average value of x'(t) over one cycle is A, so that must be the constant velocity of the hub. Since the bottom point is stationary while the velocity of the hub is A, the velocity of the top point must be 2A, which again is 4.