Question

An object's velocity is given by `v(t) = (2t^2 -2t +1 , sint )`. What is the object's rate and direction of acceleration at `t=3`?

Answer 1

`a = 10.0` `"LT"^-2`

`theta = -5.65^"o"`

Explanation:

We're asked to find the the magnitude (rate) and direction of an object's acceleration at a given time, given its velocity equation.

To do this, we need too differentiate the velocity component equations, which are given as

`v_x(t) = 2t^2 - 2t + 1`

`v_y(t) = sin(t)`

Finding the derivatives:

`a_x(t) = d/(dx) [2t^2 -2t + 1] = 4t - 2`

`a_y(t) = d/(dx)[sint] = cost` (I'll assume this is in radians)

Plugging in `t = 3` (no units), we have

`a_x = 4(3) - 2 = 10` `"LT"^-2`

`a_y = cos(3) = -0.990` `"LT"^-2`

(The `"LT"^-2` term is the dimensional form of the units for acceleration (`"distance"xx"time"^-2`). I used this term here since no units were given.)

The magnitude of the acceleration is thus

`a = sqrt((a_x)^2 + (a_y)^2) = sqrt(10^2 + (-0.990)^2) = color(red)(10.0` `color(red)("LT"^-2`

And the direction is

`theta = arctan((a_y)/(a_x)) = arctan((-0.990)/10) = color(blue)(-5.65^"o"`

Always make sure your arctangent calculation is in the right direction; it could be `180^"o"` off!