Question Type 2: Finding specific conditions on velocity and acceleration Exercises

Determine the intervals on which the object is slowing down for $t>0$.

A particle moves on a line with position $s(t)=5\sin t + t$ for $t\ge 0$. Find the first time the particle is at rest.

Find the total distance traveled by the object from $t=0.1$ to $t=\pi$ given $s(t)=e^{-t}+\frac{\sin(-t)}{t}\text{.}$

Find the first time $t>0$ at which the object is at rest, i.e. $v(t)=-e^{-t}+\frac{-t\cos t+\sin t}{t^2}=0 \text{.}$

Let $s(t)=e^{-2t}(t+1)$ for $t\ge 0$. Find all times when the acceleration is zero.

An object has velocity $v(t)=\sin t - \frac{1}{2}$ for $t\in[0,4\pi]$. Find all times in $[0,4\pi]$ when the object is decelerating (i.e., when its speed is decreasing).

Find the acceleration function $a(t)$ by differentiating your result for $v(t)=-e^{-t}+\frac{-t \cos t+\sin t}{t^2}$.

An object moves along a line with position $s(t) = t^3 - 6t^2 + 9t$ for $t \ge 0$. Find the first time $t$ when the object is at rest. [4 marks]

An object has velocity $v(t) = t^2 - 4t + 3$ for all real $t$. Determine all times $t$ when the object is speeding up.

Derive the velocity function $v(t)$ for the position $s(t)=e^{-t}+\frac{\sin(-t)}{t} \text{.}$

Determine the intervals on which the particle is speeding up.