Question Type 3: Integrating kinematic variables Exercises

A particle moves with constant acceleration $a=5 ext{ m/s}^2$, initial velocity $v(0)=2 ext{ m/s}$, and initial displacement $s(0)=0$. Find expressions for $v(t)$ and $s(t)$.

A particle has acceleration $a(t)=3t ext{ m/s}^2$ and initial velocity $v(0)=4 ext{ m/s}$. Find $v(t)$ and the distance traveled from $t=0$ to $t=4 ext{ s}$.

Given velocity $v(t)=2t^2 - t +1 ext{ m/s}$, find the acceleration $a(t)$ and the displacement between $t=1 ext{ s}$ and $t=3 ext{ s}$.

A particle has acceleration $a(t)=6t-2 ext{ m/s}^2$, with $v(1)=3 ext{ m/s}$ and $s(1)=5 ext{ m}$. Find the displacement function $s(t)$ and compute $s(4)$.

A cyclist rides so that $v(t)=5\sin t ext{ m/s}$, with $s(0)=0$. Find the net displacement from $t=0$ to $t=\pi\,$s.

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

A ball is thrown vertically upward with initial velocity $v_0=20\text{ m/s}$ from ground level. Take $a(t)=-9.8\text{ m/s}^2$. Find (a) the time to reach maximum height, (b) the maximum height reached.

Given velocity $v(t)=t^2-4t+3\text{ m/s}$, find the total distance traveled from $t=0$ to $t=5\,$s.

The displacement of a particle is given by $s(t)=t^4-4t^3+6t^2 ext{ m}$. Find its velocity $v(t)$, acceleration $a(t)$, and the times when the particle changes direction in $t\ge0$.

A car accelerates from rest with $a(t)=9-0.5t ext{ m/s}^2$. (a) Find $v(t)$ and $s(t)$. (b) Determine the time when the car comes momentarily to rest.

A particle moves with velocity $v(t)=t^3-6t^2+9t\text{ m/s}$. Determine the total distance traveled between $t=0$ and $t=3\,$s.

A particle has acceleration $a(t)=e^{-t}\text{ m/s}^2$ and initial velocity $v(0)=5\text{ m/s}$. (a) Find $v(t)$. (b) Determine the total distance traveled as $t\to\infty$.