Question Type 7: Finding the position of a specific object on a trajectory for a given value in time Exercises

An object moves along a line with velocity $v(t)=5t^2-2t+1$. If its initial position is $s(0)=4$, find its position at time $t=3$.

An object moves in the plane with velocity vector $\mathbf{v}(t)=\langle2t,3t^2\rangle$ and initial position $\mathbf{r}(0)=\langle1,-1\rangle$. Find $\mathbf{r}(2)$.

An object moves so that $v(t)=3\sin(2t)$ and $x(0)=2$. Determine $x\bigl(\frac{\pi}{2}\bigr)$.

An object has velocity $v(t)=4e^{0.5t}$ and satisfies $s(1)=0$. Find its position at time $t=4$.

Given $v(t)=t^3-6t^2+9t$ and $s(1)=5$, find $s(4)$.

The velocity of an object is $v(t)=k t^2$ and $s(0)=0$. If $s(3)=27$, determine $k$ and then compute $s(5)$.

A particle moves in the plane with $v_x=5t$, $v_y=4t^3$ and $\mathbf{r}(0)=\langle0,0\rangle$. Find $\mathbf{r}(2)$.

An object has velocity $v(t)=\frac{10}{1+t^2}$ and $s(0)=0$. Find $s(2)$.

An object moves along a line with $v(t)=50e^{-0.1t}$ and $s(0)=100$. Determine $s(10)$.

An object moves in the plane with velocity $\mathbf{v}(t)=\langle t e^t,\sin t\rangle$ and initial position $\mathbf{r}(0)=\langle0,1\rangle$. Find $\mathbf{r}(\ln2)$.

If $v(t)=3t^2\sin t$ and $s(0)=0$, find $s(\pi)$.

Find $s(2)$ if $v(t)=\ln(1+t^2)$ and $s(0)=3$.