Answer
Velocity: \[ \mathbf{v}(t) = (1 - \sin(t))\mathbf{i} + (\cos(t) - 1)\mathbf{j} \] Position: \[ \mathbf{r}(t) = \left(t + \cos(t) - 1\right)\mathbf{i} + \left(\sin(t) - t + 1\right)\mathbf{j} \]
Work Step by Step
Step 1 Given: \[ \mathbf{a}(t) = -\cos(t)\mathbf{i} - \sin(t)\mathbf{j} \] with initial conditions: \[ \mathbf{v}(0) = \mathbf{i}, \quad \mathbf{r}(0) = \mathbf{j} \] Step 2 To find the velocity: \[ \mathbf{v}(t) = \int \mathbf{a}(t) \, dt = \int (-\cos(t)\mathbf{i} - \sin(t)\mathbf{j}) \, dt \] \[ = -\sin(t)\mathbf{i} + \cos(t)\mathbf{j} + \mathbf{c} \] Since \(\mathbf{v}(0) = \mathbf{i}\), then: \[ \mathbf{c} = \mathbf{i} - \mathbf{j} \] So, \[ \mathbf{v}(t) = (1 - \sin(t))\mathbf{i} + (\cos(t) - 1)\mathbf{j} \] Step 3 To find the position: \[ \mathbf{r}(t) = \int \mathbf{v}(t) \, dt = \int ((1 - \sin(t))\mathbf{i} + (\cos(t) - 1)\mathbf{j}) \, dt \] \[ = \left(t + \cos(t) - 1\right)\mathbf{i} + \left(\sin(t) - t + 1\right)\mathbf{j} + \mathbf{c}_1 \] Since \(\mathbf{r}(0) = \mathbf{j}\), then: \[ \mathbf{c}_1 = \mathbf{j} - \mathbf{i} \] So, \[ \mathbf{r}(t) = \left(t + \cos(t) - 1\right)\mathbf{i} + \left(\sin(t) - t + 1\right)\mathbf{j} \] Result Velocity: \[ \mathbf{v}(t) = (1 - \sin(t))\mathbf{i} + (\cos(t) - 1)\mathbf{j} \] Position: \[ \mathbf{r}(t) = \left(t + \cos(t) - 1\right)\mathbf{i} + \left(\sin(t) - t + 1\right)\mathbf{j} \]
