Question Type 5: Solving basic kinematic questions using vector equations Exercises

A particle has velocity $\mathbf{v}(t)=\begin{pmatrix}6t \\ 4 \\ -2t\end{pmatrix}$ and initial position $\mathbf{r}(0)=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}$. Find the position vector $\mathbf{r}(t)$.

The position vector of an object at time $t \geq 0$ is given by $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + tk \begin{pmatrix} 5 \\ 9 \\ 1 \end{pmatrix}$.

Find the values of $k$ such that the object has a speed of $500 \text{ m s}^{-1}$.

A particle moves according to $\mathbf{r}(t) = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix}$. Find its constant speed.

A particle moves according to $\mathbf{r}(t)=\begin{pmatrix} e^t \\ t \\ \sin t \end{pmatrix}$. Find its instantaneous speed at $t=0$.

Find $k$ such that the trajectory $\mathbf{r}(t)=\begin{pmatrix} k t \\ t^2 \\ 3 \end{pmatrix}$ passes through the point $(8,4,3)$ when $t=2$.

A particle moves with $\mathbf{r}(t)=\begin{pmatrix} 5\cos t \\\\ 5\sin t \\\\ 3t \end{pmatrix}$. Determine its speed as a function of $t$.

The position of a particle is given by $\mathbf{r}(t) = \begin{pmatrix} t^2 \\ 3t \\ 4 \end{pmatrix}$. Find the velocity vector and the speed at $t=2$.

A particle moves on the line $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t\begin{pmatrix} 4 \\ -2 \\ 1 \end{pmatrix}$.

Find the values of $t$ for which the particle is $10$ units from the origin.

The path of a particle is $\mathbf r(t) = \begin{pmatrix} t \\ t^2 \\ t^3 \end{pmatrix}$. Find its acceleration vector at $t=2$.

The path of a particle is given by $\mathbf{r}(t) = \begin{pmatrix} t \\ t^3 \\ t^2 \end{pmatrix}$, where $t$ denotes time.

Find the value of $t$ for which the speed of the particle is $10$.

A particle has constant acceleration $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and initial velocity $\mathbf{v}(0) = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$. Find $\mathbf{v}(t)$ and its speed at $t=4$.

A projectile is launched from the origin with initial velocity $\begin{pmatrix} 10 \\ 10 \\ 0 \end{pmatrix}$ and acceleration $\begin{pmatrix} 0 \\ 0 \\ -9.8 \end{pmatrix}$. Its position is given by $\mathbf{r}(t) = \begin{pmatrix} 10t \\ 10t \\ 0 \end{pmatrix} + \frac{1}{2}t^2 \begin{pmatrix} 0 \\ 0 \\ -9.8 \end{pmatrix}.$ Find its speed at $t=1$.