IB Mathematics Applications & Interpretation Question Type 1: Deriving to Find the Rate of Change of a Kinematic Variable Exercises

The acceleration of a particle is given by $a(t)=12t^2-6t+1$. At $t=1$, the velocity $v(1)=2$ and the displacement $s(1)=5$. Find expressions for $v(t)$ and $s(t)$.

A particle's displacement is given by $s(t)=4t-t^2+3e^t$. Find the time $t$ when the acceleration $a(t)$ is zero.

Given the displacement function $$s(t)=e^{-t}+\frac{\sin (-t)}{t},$$ for $t>0$, find the acceleration $a(t)$.

For $s(t)=t{e}^{-t}$, find expressions for $v(t)$ and $a(t)$.

Given $v(t)=5e^{-2t}$ and $s(0)=0$, find the displacement $s(t)$.

The acceleration of a particle is $a(t)=6t-4$ for $t\ge 0$. Given that $v(0)=3$ and $s(0)=2$, find expressions for the velocity $v(t)$ and the displacement $s(t)$.

Given the velocity of a particle $$v(t)=t^3-3t^2+t+10,$$ where $t$ is measured in seconds and $v$ in metres per second, find the total distance travelled by the particle between $t=0$ and $t=5$.

A particle is thrown vertically upward with initial velocity $v(0)=20\text{m}{\text{s}}^{-1}$ under constant acceleration $a=-9.8\text{m}{\text{s}}^{-2}$. Given $s(0)=0$, find the time when it returns to $s=0$.

Find the first time $t>0$ when the object is at rest. Give your answer to two decimal places.

The velocity of a particle moving along a straight line is given by $v(t)=\cos (2t)-2\sin t$, for $t\ge 0$, where $t$ is the time in seconds.

Given that the displacement $s$ is $1$ metre when $t=0$, find an expression for $s(t)$ in terms of $t$.

An object moves with displacement $$s(t)=\ln (1+t)-t.$$ Find the time $t>0$ when it changes direction.

Given $s(t)=3\sin t-t^2$, find $v(t)$, $a(t)$, and the smallest time when $a(t)=0$ for $t>0$.