Question

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`?

Answer 1

`2.0 "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 = dx/dt`

The derivative of a constant is `0`, and the derivative of `t^n` is `nt^(n-1)`. Also, the derivative of `sin (at)` is `acos(ax)`. Using these formulas, the differentiation of the position equation is

`v_x(t) = 2 - pi/4 cos(pi/8 t)`

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"`