Question

Find

(a) the displacement vectors from r(a) tor(b),

(b) the magnitude of the displacement vector, and

(c) the distance travelled by a particle on the curve from a to b.

r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Short answer

Step-by-step solution

Part (a) Step 1. Given Information

The given function is r(t) = α sinβt,α cosβt,γt, a = 0, b = 1

Part (a) Step 2. Finding the  displacement vector   

Part (b) Step 1.  Finding the magnitude of the displacement vector,  

Part (c) Step 1. Finding the  displacement vector  

$$\begin{gathered} r\left(t\right)=⟨\alpha \sin \pi t,\alpha \cos \beta t,\gamma t⟩ \\ ||r^{\text{'}}\left(t\right)||=\sqrt{{\alpha }^2{\beta }^2{\cos }^2\beta t+{\alpha }^2{\beta }^2{\sin }^2\beta t+{\gamma }^2} \\ =\sqrt{{\alpha }^2{\beta }^2\left({\cos }^2\beta t+{\sin }^2\beta t\right)+{\gamma }^2} \\ =\sqrt{{\alpha }^2{\beta }^2+{\gamma }^2} \\ \mathrm{The}\mathrm{distance}\mathrm{travelled}\mathrm{from}\mathrm{a}\mathrm{to}\mathrm{b}\mathrm{is} \\ ={\int }_0^1\sqrt{{\alpha }^2{\beta }^2+{\gamma }^2}dt \\ =\left(\sqrt{{\alpha }^2{\beta }^2+{\gamma }^2}\right)(1-0)\text{substitute the limits for}t \\ =\sqrt{{\alpha }^2{\beta }^2+{\gamma }^2} \\ \mathrm{The}\mathrm{distance}\mathrm{travelled}\mathrm{from}\mathrm{a}\mathrm{to}\mathrm{b}\mathrm{is} \\ =\sqrt{{\alpha }^2{\beta }^2+{\gamma }^2} \end{gathered}$$