Mathematics, 27.11.2019 05:31 sassyunicorngir

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=

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Answer from: khynia11

The particle moves at $\sqrt{\sin^2(t) + \frac{\cos^2(\frac{t}{2})}{4}$ feets per second after t seconds.The first stop is at π secondsThe particle stops at nπ seconds, with n an odd integer

Step-by-step explanation:

We can write the position with a function $f: R \rightarrow R^2$ , f(t) = (cos(t), sin(t/2)) .

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

$f'(t) = (-\sin(t), \cos(t/2) * 1/2 )$

So, the speed of the particle after t seconds is || f'(t) || = $\sqrt{(-\sin(t))^2 + (\cos(\frac{t}{2})*\frac{1}{2})^2} = \sqrt{\sin^2(t) + \frac{\cos^2(\frac{t}{2})}{4}$ 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:

if t = nπ, with n odd, not only cos(t/2) = 0, but also sin(t) = 0The first positive value for which both sin(t) and cos(t/2) are 0 is nπ, with n equal to 1, thus, t = π.

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.

Answer from: Quest

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents

Answer the following and be the brainiest the mistake in the pattern is that : a. the decimal point

Answer from: Quest

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove.

an inaccurate proof leads to errors, or stating results that are not accurate. which wil mess the probelm up you dont want this

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Aparticle moves with its position given by x = cos(t) and y = sin (t/2 ), where positions are given...