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

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

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

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

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

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

Consider $s(t) = t \ln t - t$, for $t > 0$, where $s$ is the displacement of a particle at time $t$.

Find the values of $t$ at which the speed of the particle is equal to $1$.

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

An object moves along a straight line such that its displacement, $s$ metres, from a fixed point $O$ at time $t$ seconds, $t \ge 0$, is given by $s(t) = t^4 - 4t^3 + 6t^2$.

Determine the times, if any, when the object changes direction.

Let $s(t) = \frac{1}{3}t^3 - t$ be the displacement of a particle at time $t$. Find the time at which the velocity attains its minimum value.

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

An object moves with displacement $s(t) = e^{-t} \cos t$. Find the first positive value of $t$ for which the acceleration is zero.

Let $s(t) = \sqrt{t}, \quad t \ge 0$. Determine for which $t$ the acceleration is decreasing.