2. Time Lord (#differentiation) Have you ever noticed in a movie that the wheels of a car will seem to spin in the opposite direction that you would expect? That is due to what is called the stroboscopic effect. Consider a wheel of radius 1 m. Let's say that at time t=0, we paint a dot on the wheel at its right most point. If the wheel takes 7 seconds to complete a full revolution, then the x-coordinate of our painted dot would be given by the function r = cos (t), and the y-coordinate y = sin(t). (a) You are a photographer taking one photo every 7 seconds, make a table showing the x- and y- coordinates of the dot in your first four photos. What is the average rate of change of each coordinate between consecutive photos? Interpret the units (b) You now change your rate, taking one photo every 6 seconds. Make a new table and calculate the average rate of change between consecutive shots for your first four photos. (c) What is the instantaneous rate of change of the x- and y- coordinates at t = 0? How does that compare with the average rate you computed on part (b)? If you just looked at the photos, would you say the wheels are spinning faster or slower than they really are? (d) Sketch two graphs: one for the x-coordinate and another for the y-coordinate as a function of time as continuous blue curves for 50 seconds. Use red points to indicate the r- and y-coordinates of when photos were taken every six seconds. If you connect the dots, how does that function relate to the original function?
2. Time Lord (#differentiation) Have you ever noticed in a movie that the wheels of a car will seem to spin in the opposite direction that you would expect? That is due to what is called the stroboscopic effect. Consider a wheel of radius 1 m. Let's say that at time t=0, we paint a dot on the wheel at its right most point. If the wheel takes 7 seconds to complete a full revolution, then the x-coordinate of our painted dot would be given by the function r = cos (t), and the y-coordinate y = sin(t). (a) You are a photographer taking one photo every 7 seconds, make a table showing the x- and y- coordinates of the dot in your first four photos. What is the average rate of change of each coordinate between consecutive photos? Interpret the units (b) You now change your rate, taking one photo every 6 seconds. Make a new table and calculate the average rate of change between consecutive shots for your first four photos. (c) What is the instantaneous rate of change of the x- and y- coordinates at t = 0? How does that compare with the average rate you computed on part (b)? If you just looked at the photos, would you say the wheels are spinning faster or slower than they really are? (d) Sketch two graphs: one for the x-coordinate and another for the y-coordinate as a function of time as continuous blue curves for 50 seconds. Use red points to indicate the r- and y-coordinates of when photos were taken every six seconds. If you connect the dots, how does that function relate to the original function?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Part C & D needed correctly
Transcribed Image Text:2. Time Lord (#differentiation)
Have you ever noticed in a movie that the wheels of a car will seem to spin in
the opposite direction that you would expect? That is due to what is called the
stroboscopic effect. Consider a wheel of radius 1 m. Let's say that at time t=0,
we paint a dot on the wheel at its right most point. If the wheel takes 7 seconds to
complete a full revolution, then the x-coordinate of our painted dot would be given
by the function = cos (t), and the y-coordinate y = sin(t).
(a) You are a photographer taking one photo every 7 seconds, make a table showing
the x- and y- coordinates of the dot in your first four photos. What is the average
rate of change of each coordinate between consecutive photos? Interpret the units
(b) You now change your rate, taking one photo every 6 seconds. Make a new table
and calculate the average rate of change between consecutive shots for your first
four photos.
(c) What is the instantaneous rate of change of the x- and y- coordinates at t = 0?
How does that compare with the average rate you computed on part (b)? If you
just looked at the photos, would you say the wheels are spinning faster or slower
than they really are?
(d) Sketch two graphs: one for the x-coordinate and another for the y-coordinate as
a function of time as continuous blue curves for 50 seconds. Use red points to
indicate the x- and y-coordinates of when photos were taken every six seconds.
If you connect the dots, how does that function relate to the original function?
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