Question

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`?

Answer 1

`"speed" = 8.94` `"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(t) = d/(dt) [2t - 2tsin(pi/4t) + 2]`

`= 2 - pi/2tcos(pi/4t)`

The speed at `t = 7` `"s"` is found by

`v(7) = 2 - pi/2(7)cos(pi/4(7))`

`= 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.