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 $t \geq 0$. 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 $t > 0$.