An object is moving in the plane with parametric equations: x(t) = cos(πt), y(t) = 3sin(πt). The path traced out is an ellipse, as pictured. We assume t is non-negative and in units of "seconds". Assume units on the axes are "feet". The location of the object at time t is P(t) = (cos(πt),3sin(πt)). The picture below is the derivative of the speed of the object at time t. I already found the speed of the object at time t and took the derivative. I know that you need the derivative to find the values but I do not know where to start.
How do I find the maximum and minimum speed?
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calculus
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1set acceleration equal to 0 and solve for t. plug this t value in for velocity which you have found – 2017-02-06
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0Not the acceleration, the derivative of the speed (which is not the same thing for motion that isn't in a straight line). – 2017-02-06
1 Answers
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Consider the squared velocity (which is more algebraically tractable than the actual velocity): $$x'(t)^2 + y'(t)^2 = \pi^2 \sin^2 (\pi t) + 9 \pi^2 \cos^2 (\pi t) = \pi^2 (1 + 8 \cos^2 (\pi t)).$$
The maxima of this are wherever $\cos^2 (\pi t) = 1$. You should be able to take it the rest of the way.