The position of a particle is given by $s = 5 \cos (2t+ (\pi/4))$ at time $t$ . What are the maximum values of the displacement,the velocity,and the acceleration? The answers are displacement: $5$ velocity: $10$ acceleration: $20$
I tried to do it but I am not getting the right answer. Here is my solution:
$s(t) = 5\cos(2t+ \pi/4)$
$s'(t) = -10 \sin(2t + \pi/4)$
$0= -10[\sin2t\cos\pi/4 + \cos2t\sin\pi/4]$
$-\sin2t(\sqrt{2}/2) = \cos2t(\sqrt{2}/2)$
$-1=\tan2t$
$\tan2t = 3\pi/8, 7\pi/8$
when I substitute the values, I get -5. For the v(t).I got zero. please help thanks!
The displacement is $5\cos\left(2t+\frac{\pi}{4}\right)$. Use what you know about the geometry of the cosine curve.
The biggest a cosine ever gets is $1$. That will happen in our case, for example, when $2t+\frac{\pi}{4}=2\pi$. So the biggest displacement gets is $5$.
Instead, you differentiated and set the result equal to $0$. That's OK, we will need the derivative in order to talk about velocity.
The derivative turns out to be $-10\sin\left(2t+\frac{\pi}{4}\right)$. You wanted to find out where this is $0$, and used the formula for the sine of a sum. It is much easier to note that the derivative is $0$ when $2t+\frac{\pi}{4}$ is a multiple of $\pi$. But examining the derivative to find the maximum is not the best procedure here.
To finish the problems, calculate velocity (you already did) and acceleration. Using the simple method I used to find the max displacement, find the max velocity, and the max acceleration.