Aparticle moves with its position given by x = cos(t) and y = sin (t/2 ), where positions are given in feet from the origin and time t is in seconds. find the speed of the particle. speed = (include units) find the first positive time when the particle comes to a stop. (include units) t =
if n is any odd integer, write a formula (in terms of n) for all positive times t at which the particle comes to a stop. (include units) t=

Step-by-step explanation:

We can write the position with a function ,

In order to find the speed, we have to find the norm of the derivate of f. The derivate of cos(x) is -sin(x), and the derivate of sin(x) is cos(x). Accoring to the chain rule, in order to derivate sin(t/2), we have to multiply cos(t/2) by the derivate of t/2, 1/2. Thus the derivate of sin(t/2) is cos(t/2) * 1/2. Therefore, we have

So, the speed of the particle after t seconds is || f'(t) || = feats per second.

The particle stops when f'(t) = (-sin(t),cos(t/2) * 1/2) = (0,0). That can only happpen if sin(t) = cos(t/2) = 0. The values where sin(t) is equal to 0 follow the expression kπ, where k in an integer. The values where cos(t) = 0 have the form π/2 + jπ, with j another integer. Thus, cos(t/2) = 0 only when t = 2 * (π/2 + jπ) = π + 2jπ. In other words, t = nπ, with n an odd integer. In order to conclude the excercise we have to note a few things:

With this information, we can conclude that the first positive time the particle stops is at π seconds, and the particle stops at nπ seconds, with n being any odd integer.

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step-by-step explanation: