Given: \[ \mathbf{r}(t) = (-2t \sin t,\;-2t \cos t,\;-9t^2) \] Let's find the **velocity**, **acceleration**, and **speed**. --- ## 1. **Velocity** (Derivative of \(\mathbf{r}(t)\)) \[ \mathbf{v}(t) = \frac{d\mathbf{r}}{dt} \] Differentiate each component: - \(x(t) = -2t \sin t\) - \(y(t) = -2t \cos t\) - \(z(t) = -9t^2\) ### \(x(t)\): \[ \frac{d}{dt}(-2t \sin t) = -2 \frac{d}{dt}(t \sin t) = -2 (\sin t + t \cos t) \] (by product rule: \(\frac{d}{dt}[uv] = u'v + uv'\)) ### \(y(t)\): \[ \frac{d}{dt}(-2t \cos t) = -2 \frac{d}{dt}(t \cos t) = -2 (\cos t - t \sin t) \] (\(\frac{d}{dt}[t \cos t] = \cos t - t \sin t\)) ### \(z(t)\): \[ \frac{d}{dt}(-9t^2) = -18t \] So, \[ \boxed{ \mathbf{v}(t) = \left( -2(\sin t + t \cos t),\; -2(\cos t - t \sin t),\; -18t \right) } \] --- ## 2. **Acceleration** (Derivative of velocity) Differentiate each component of \(\mathbf{v}(t)\): ### First component: \[ -2(\sin t + t \cos t) \] Derivative: \[ -2(\cos t + [\cos t - t \sin t]) = -2(\cos t + \cos t - t \sin t) = -2(2\cos t - t \sin t) = -4\cos t + 2t \sin t \] ### Second component: \[ -2(\cos t - t \sin t) \] Derivative: \[ -2(-\sin t - [\sin t + t \cos t]) = -2(-\sin t - \sin t - t \cos t) = -2(-2\sin t - t \cos t) = 4\sin t + 2t \cos t \] ### Third component: \[ -18t \implies -18 \] So, \[ \boxed{ \mathbf{a}(t) = \left( -4\cos t + 2t \sin t,\; 4\sin t + 2t \cos t,\; -18 \right) } \] --- ## 3. **Speed** (Magnitude of velocity) \[ \text{Speed} = \|\mathbf{v}(t)\| = \sqrt{v_x^2 + v_y^2 + v_z^2} \] Plug in: - \(v_x = -2(\sin t + t \cos t)\) - \(v_y = -2(\cos t - t \sin t)\) - \(v_z = -18t\) Compute: \[ \begin{align*} \text{Speed} &= \sqrt{ \left[ -2(\sin t + t \cos t) \right]^2 + \left[ -2(\cos t - t \sin t) \right]^2 + (-18t)^2 } \\ &= \sqrt{ 4(\sin t + t \cos t)^2 + 4(\cos t - t \sin t)^2 + 324 t^2 } \end{align*} \] Now expand the first two terms: \[ (\sin t + t \cos t)^2 = \sin^2 t + 2 t \sin t \cos t + t^2 \cos^2 t \] \[ (\cos t - t \sin t)^2 = \cos^2 t - 2 t \sin t \cos t + t^2 \sin^2 t \] Add together: \[ (\sin t + t \cos t)^2 + (\cos t - t \sin t)^2 = \sin^2 t + t^2 \cos^2 t + \cos^2 t + t^2 \sin^2 t \] Sum up terms: - \(\sin^2 t + \cos^2 t = 1\) - \(t^2 (\sin^2 t + \cos^2 t) = t^2\) So, \[ (\sin t + t \cos t)^2 + (\cos t - t \sin t)^2 = 1 + t^2 \] Therefore, \[ \text{Speed} = \sqrt{4(1 + t^2) + 324 t^2} = \sqrt{4 + 4 t^2 + 324 t^2} = \sqrt{4 + 328 t^2} \] \[ \boxed{ \text{Speed} = \sqrt{4 + 328 t^2} } \] --- ### **Summary** \[ \boxed{ \begin{aligned} \mathbf{v}(t) &= \big( -2(\sin t + t \cos t),\; -2(\cos t - t \sin t),\; -18t \big) \\[1em] \mathbf{a}(t) &= \big( -4\cos t + 2t \sin t,\; 4\sin t + 2t \cos t,\; -18 \big) \\[1em] \text{Speed} &= \sqrt{4 + 328 t^2} \end{aligned} } \]