Question Type 2: Finding specific conditions on the rate of change of a kinematic variable Exercises

Let $s(t)=\sqrt{t},\quad t\ge0.$ Determine for which $t$ the acceleration is decreasing.

For $s(t)=5t-\frac{1}{t},\quad t>0,$ find the intervals on which the acceleration is positive.

Given $s(t)=t^3-6t^2+9t,$ determine the intervals on which the object is moving to the right (i.e./ $v(t)>0$).

Given $s(t)=e^{2t}-4t,$ find all times $t$ such that the object is instantaneously at rest.

Let $s(t)=\tfrac{1}{3}t^3 - t.$ Find the time at which the velocity attains its minimum value.

Let $s(t)=\ln t + t^{-1},\quad t>0.$ Find the inflection points of the motion (i.e./ where $a'(t)=0$).

An object moves according to $s(t)=t^4-4t^3+6t^2.$ Determine the times when the object changes direction.

For the motion $s(t)=2\sin t - t,$ find the times when the acceleration is zero.

Consider $s(t)=t\ln t - t,\quad t>0.$ Find the time at which the speed equals 1.

For $s(t)=\tfrac{t^2}{2}+3\ln t,\quad t>0,$ find the time at which the speed is minimized.

An object moves with $s(t)=e^{-t}\cos t.$ Find the first positive time when the acceleration is zero.

For the position function s(t)=e^{-t}+rac{\\sin(-t)}{t},\quad t>0, find the first positive time $t$ at which the object is at rest.