Question

A decomposition of the acceleration vector: Find ${\mathrm{comp}}_{\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}}\mathbf{a}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}$ where v anda are the velocity and acceleration vectors, respectively, of the following functions.

Short answer

The component of a(t)alongv(t)is$com{p}_{v\left(t\right)}a\left(t\right)=\sin \sqrt{2}t.$

Step-by-step solution

Step 1. Given Information.

The given function is

Step 2. Calculating the velocity and acceleration vector. 

Let's calculate the given function,

Step 3. Solve.

Put (a) and (b) in $com{p}_{v\left(t\right)}a\left(t\right)=\frac{\left|a\left(t\right).v\left(t\right)\right|}{||v\left(t\right)||},$

$$\begin{gathered} com{p}_{v\left(t\right)}a\left(t\right)=\frac{\left|2\sin t\cos t\right|}{\sqrt{2}} \\ com{p}_{v\left(t\right)}a\left(t\right)=\frac{2\sin t\cos t}{\sqrt{2}} \\ com{p}_{v\left(t\right)}a\left(t\right)=\sqrt{2}\sin t\cos t \\ com{p}_{v\left(t\right)}a\left(t\right)=\sin \sqrt{2}t \end{gathered}$$