Question

find the velocity, acceleration, and speed of a particle with position function r(t) = <8t sin t, 8t cos t, 5t^2>. v(t) = <8 sin t + 8t cos t, 8 cos t - 8t sin t, 10t> a(t) = <8 cos t + 8 cos t - 8t sin t, -8 sin t - 8 sin t - 8t cos t, 10> |v(t)|=

Explanation:

Step1: Recall velocity - acceleration - position relationship

The velocity vector $\mathbf{v}(t)$ is the derivative of the position vector $\mathbf{r}(t)$, and the acceleration vector $\mathbf{a}(t)$ is the derivative of the velocity vector $\mathbf{v}(t)$. Given $\mathbf{r}(t)=\langle8t\sin t,8t\cos t,5t^{2}
angle$.

Step2: Differentiate $\mathbf{r}(t)$ to get $\mathbf{v}(t)$

Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where if $u = 8t$ and $v=\sin t$ for the first component, $(8t\sin t)^\prime=8\sin t + 8t\cos t$. For the second component, if $u = 8t$ and $v=\cos t$, $(8t\cos t)^\prime=8\cos t-8t\sin t$. For the third component, $(5t^{2})^\prime = 10t$. So $\mathbf{v}(t)=\langle8\sin t + 8t\cos t,8\cos t-8t\sin t,10t
angle$.

Step3: Differentiate $\mathbf{v}(t)$ to get $\mathbf{a}(t)$

Differentiating the first component of $\mathbf{v}(t)$: $(8\sin t + 8t\cos t)^\prime=8\cos t+8\cos t - 8t\sin t=16\cos t-8t\sin t$. Differentiating the second component: $(8\cos t-8t\sin t)^\prime=-8\sin t-8\sin t-8t\cos t=-16\sin t - 8t\cos t$. Differentiating the third component: $(10t)^\prime = 10$. So $\mathbf{a}(t)=\langle16\cos t-8t\sin t,-16\sin t - 8t\cos t,10
angle$.

Step4: Calculate the speed $|\mathbf{v}(t)|$

The speed is the magnitude of the velocity vector. $|\mathbf{v}(t)|=\sqrt{(8\sin t + 8t\cos t)^{2}+(8\cos t-8t\sin t)^{2}+(10t)^{2}}$. Expand the first two terms: $(8\sin t + 8t\cos t)^{2}=64\sin^{2}t + 128t\sin t\cos t+64t^{2}\cos^{2}t$ and $(8\cos t-8t\sin t)^{2}=64\cos^{2}t-128t\sin t\cos t + 64t^{2}\sin^{2}t$. Then $(8\sin t + 8t\cos t)^{2}+(8\cos t-8t\sin t)^{2}=64(\sin^{2}t+\cos^{2}t)+64t^{2}(\sin^{2}t+\cos^{2}t)=64 + 64t^{2}$. And $(10t)^{2}=100t^{2}$. So $|\mathbf{v}(t)|=\sqrt{64 + 64t^{2}+100t^{2}}=\sqrt{64 + 164t^{2}}$.

Answer:

$\sqrt{64 + 164t^{2}}$