# The position of an object moving along a line is given by p(t) = 2t - 2tsin(( pi )/4t) + 2 . What is the speed of the object at t = 7 ?

Jul 12, 2017

$\text{speed} = 8.94$ $\text{m/s}$

#### Explanation:

We're asked to find the speed of an object with a known position equation (one-dimensional).

To do this, we need to find the velocity of the object as a function of time, by differentiating the position equation:

$v \left(t\right) = \frac{d}{\mathrm{dt}} \left[2 t - 2 t \sin \left(\frac{\pi}{4} t\right) + 2\right]$

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

The speed at $t = 7$ $\text{s}$ is found by

$v \left(7\right) = 2 - \frac{\pi}{2} \left(7\right) \cos \left(\frac{\pi}{4} \left(7\right)\right)$

= color(red)(-8.94 color(red)("m/s" (assuming position is in meters and time in seconds)

The speed of the object is the magnitude (absolute value) of this, which is

"speed" = |-8.94color(white)(l)"m/s"| = color(red)(8.94 color(red)("m/s"

The negative sign on the velocity indicates that the particle is traveling in the negative $x$-direction at that time.