Question

the equation below gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). find the bodys velocity, speed, acceleration, and jerk at time t = π/4 sec. s = 5 - 3 cos t the velocity of the body at t = π/4 sec is m/sec.

Explanation:

Step1: Recall velocity - position relationship

Velocity $v(t)$ is the derivative of position $s(t)$. Given $s(t)=5 - 3\cos t$, by the derivative formula $\frac{d}{dt}(\cos t)=-\sin t$ and $\frac{d}{dt}(C) = 0$ (where $C$ is a constant), we have $v(t)=\frac{ds}{dt}=3\sin t$.

Step2: Evaluate velocity at $t = \frac{\pi}{4}$

Substitute $t=\frac{\pi}{4}$ into $v(t)$. We know that $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. So $v(\frac{\pi}{4})=3\sin\frac{\pi}{4}=3\times\frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{2}$.

Answer:

$\frac{3\sqrt{2}}{2}$