# The position of an object moving along a line is given by p(t) = 2t - 2sin(( pi )/8t) + 2 . What is the speed of the object at t = 12 ?

May 24, 2017

$2.0 \text{m"/"s}$

#### Explanation:

We're asked to find the instantaneous $x$-velocity ${v}_{x}$ at a time $t = 12$ given the equation for how its position varies with time.

The equation for instantaneous $x$-velocity can be derived from the position equation; velocity is the derivative of position with respect to time:

${v}_{x} = \frac{\mathrm{dx}}{\mathrm{dt}}$

The derivative of a constant is $0$, and the derivative of ${t}^{n}$ is $n {t}^{n - 1}$. Also, the derivative of $\sin \left(a t\right)$ is $a \cos \left(a x\right)$. Using these formulas, the differentiation of the position equation is

${v}_{x} \left(t\right) = 2 - \frac{\pi}{4} \cos \left(\frac{\pi}{8} t\right)$

Now, let's plug in the time $t = 12$ into the equation to find the velocity at that time:

v_x(12"s") = 2 - pi/4 cos(pi/8 (12"s")) = color(red)(2.0 "m"/"s"