### Particle Motion Analysis#### Problem StatementA particle initially located at the origin has an acceleration of \(3.00 \, \mathbf{j} \, \text{m/s}^2\) and an initial velocity of \(5.00 \, \mathbf{i} \, \text{m/s}\). Find:a) The vector position of the particle at any time b) The velocity of the particle at any time c) The coordinates of the particle at \(t = 2.00 \, \text{s}\) d) The speed of the particle at \(t = 2.00 \, \text{s}\) #### Solution Approach**a) The vector position of the particle at any time**To find the vector position \(\mathbf{r}(t)\) of the particle, we need to integrate the velocity function \(\mathbf{v}(t)\). Given that the initial velocity is \( \mathbf{v}(0) = 5.00 \, \mathbf{i} \, \text{m/s} \) and the acceleration \( \mathbf{a} = 3.00 \, \mathbf{j} \, \text{m/s}^2 \):\[ \mathbf{v}(t) = \mathbf{v}(0) + \mathbf{a}t \]\[ \mathbf{v}(t) = 5.00 \, \mathbf{i} + 3.00 \, \mathbf{j}t \]Now, integrating the velocity to find the position:\[ \mathbf{r}(t) = \int \mathbf{v}(t) \, dt \]\[ \mathbf{r}(t) = \int (5.00 \, \mathbf{i} + 3.00 \, \mathbf{j}t) \, dt \]\[ \mathbf{r}(t) = (5.00 \, \mathbf{i})t + \left( \frac{3.00}{2} \, \mathbf{j} \right)t^2 + \mathbf{C} \]Since the particle starts at the origin, \(\mathbf{r}(0) = 0\):\[ \mathbf{C} = 0 \]Thus, the position vector:\[ \mathbf{r}(t) = 5.00t \, \mathbf{i}

Name: ### Particle Motion Analysis#### Problem StatementA particle initiall
Uploaded: 2024-03-20T06:57:47.000Z
Channel: Bartleby
Description: ### Particle Motion Analysis#### Problem StatementA particle initially located at the origin has an acceleration of \(3.00 \, \mathbf{j} \, \text{m/s}^2\) and an initial velocity of \(5.00 \, \mathbf{i} \, \text{m/s}\). Find:a) The vector position o