Question Type 3: Going from acceleration and velocity, to velocity and displacement Exercises

Given the acceleration function $a(t)=4 \, \text{m}\,\text{s}^{-2}$, with initial velocity $v(0)=2 \, \text{m}\,\text{s}^{-1}$ and initial displacement $s(0)=1 \, \text{m}$, find the displacement function $s(t)$.

Given $v(t)=\dfrac{5}{t+1}$ m/s, find the displacement from $t=0$ to $t=4$.

A particle has acceleration $a(t)=2\sin t \text{ m/s}^2$ and initial conditions $v(0)=0$ and $s(0)=0$.

Find the displacement of the particle over the interval $[0,\pi]$.

A ball is thrown upward so that $a(t)=-9.8 \, \text{m/s}^2$, $v(0)=20 \, \text{m/s}$ and $s(0)=0$. Determine its maximum height.

Given $a(t)=6t-4 \, \text{m\,s}^{-2}$, $v(1)=5 \, \text{m\,s}^{-1}$ and $s(1)=2 \, \text{m}$, find the displacement function $s(t)$.

If $v(t)=4\cos(2t)$ m/s, find the displacement from $t=0$ to $t=\pi/2$. [3 marks]

A particle moves with velocity $v(t)=t^3-3t^2+t+4$ m/s. Find its total distance traveled between $t=0$ and $t=2$.

Find the displacement of an object on $[0,3]$ if its velocity is $v(t)=3t^2-2t+1$ m/s.

An object has acceleration $a(t)=6t-4$ $\text{m}\,\text{s}^{-2}$. Given $v(1)=5$ $\text{m}\,\text{s}^{-1}$, find the velocity function $v(t)$.

An object with $v(t)=5t^2$ m/s starts from rest at $s(0)=0$. Find $s(t)$.