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

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

An object moves along a line such that its velocity $v$ at time $t$ is given by $v(t) = 50e^{-0.1t}$ for $t ≥ 0$. The initial displacement of the object is $s(0) = 100$.

Determine the displacement of the object when $t = 10$.

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 $t = 3$.

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 $\mathbf{r}(0) = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$. Find $\mathbf{r}(2)$.

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

Given that $v(t) = 3t^2 \sin t$ and $s(0) = 0$, find the value of $s(\pi)$.

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

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

Given the velocity function $v(t) = t^3 - 6t^2 + 9t$ and the initial condition $s(1) = 5$, find the position $s(4)$.

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

An object moves in the plane with velocity $\mathbf{v}(t) = \begin{pmatrix} t e^t \\ \sin t \end{pmatrix}$ and initial position $\mathbf{r}(0) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Find $\mathbf{r}(\ln 2)$.