IB Mathematics Analysis and Approaches 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.