Question

Problem Zero: Read the section and make your own summary of the material.

Short answer

MOTION:

$$\begin{gathered} 1.\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{V}\mathit{e}\mathit{r}\mathit{s}\mathit{u}\mathit{s}\mathbf{}\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{T}\mathit{r}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{d}\mathbf{:} \\ Letthevector-valuedfunction\mathit{r}\left(t\right)representthepositionfunctionfora \\ particletravellingonacurveC.Ifthedomainof\mathit{r}isanintervalIcontaining \\ pointsaandb: \\ \left(a\right)The\mathit{d}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}ofthetheparticlefromatobisgivenbythevector\mathit{r}\left(b\right)-\mathit{r}\left(a\right). \\ \left(b\right)The\mathit{d}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{t}\mathit{r}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{d}bytheparticlefromatobisgivenbythescalar{\int }_a^b\left|\left|\mathit{r}\text{'}\left(t\right)\right|\right|dt. \end{gathered}$$

$$\begin{gathered} 2.\mathit{T}\mathit{h}\mathit{e}\mathbf{}\mathit{T}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{l}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{N}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathbf{}\mathit{C}\mathit{o}\mathit{m}\mathit{p}\mathit{o}\mathit{n}\mathit{e}\mathit{n}\mathit{t}\mathit{s}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{A}\mathit{c}\mathit{c}\mathit{e}\mathit{l}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{:} \\ Let\mathit{r}\left(t\right)beatwice-differentiablevectorfunctionwitheithertwoorthreecomponents \\ andwithderivatives\mathit{r}\mathbf{\text{'}}\left(t\right)=\mathit{v}\left(t\right)and\mathit{r}\mathbf{\text{'}\text{'}}\left(t\right)=\mathit{a}\left(t\right).Thenthe\tan gentialcomponentof \\ acceleration{a}_{T}isgivenby \\ {a}_{\mathit{T}}=\frac{\mathit{v}\mathbf{.}\mathit{a}}{\left|\left|\mathit{v}\right|\right|} \\ andthenormalcomponentofacceleration{a}_{\mathit{N}}isgivenby \\ {a}_{\mathit{N}}=\frac{\left|\left|\mathit{v}\mathbf{\times }\mathit{a}\right|\right|}{\left|\left|\mathit{v}\right|\right|}. \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

Given is the text on the topic Motion.

Step 2. Summary of the text given.

MOTION:

$$\begin{gathered} 1.\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{V}\mathit{e}\mathit{r}\mathit{s}\mathit{u}\mathit{s}\mathbf{}\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{T}\mathit{r}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{d}\mathbf{:} \\ Letthevector-valuedfunction\mathit{r}\left(t\right)representthepositionfunctionfora \\ particletravellingonacurveC.Ifthedomainof\mathit{r}isanintervalIcontaining \\ pointsaandb: \\ \left(a\right)The\mathit{d}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}ofthetheparticlefromatobisgivenbythevector\mathit{r}\left(b\right)-\mathit{r}\left(a\right). \\ \left(b\right)The\mathit{d}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{t}\mathit{r}\mathit{a}\mathit{v}\mathit{e}\mathit{l}\mathit{l}\mathit{e}\mathit{d}bytheparticlefromatobisgivenbythescalar{\int }_a^b\left|\left|\mathit{r}\text{'}\left(t\right)\right|\right|dt. \end{gathered}$$

$$\begin{gathered} 2.\mathit{T}\mathit{h}\mathit{e}\mathbf{}\mathit{T}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{l}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{N}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathbf{}\mathit{C}\mathit{o}\mathit{m}\mathit{p}\mathit{o}\mathit{n}\mathit{e}\mathit{n}\mathit{t}\mathit{s}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{A}\mathit{c}\mathit{c}\mathit{e}\mathit{l}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{:} \\ Let\mathit{r}\left(t\right)beatwice-differentiablevectorfunctionwitheithertwoorthreecomponents \\ andwithderivatives\mathit{r}\mathbf{\text{'}}\left(t\right)=\mathit{v}\left(t\right)and\mathit{r}\mathbf{\text{'}\text{'}}\left(t\right)=\mathit{a}\left(t\right).Thenthe\tan gentialcomponentof \\ acceleration{a}_{T}isgivenby \\ {a}_{\mathit{T}}=\frac{\mathit{v}\mathbf{.}\mathit{a}}{\left|\left|\mathit{v}\right|\right|} \\ andthenormalcomponentofacceleration{a}_{\mathit{N}}isgivenby \\ {a}_{\mathit{N}}=\frac{\left|\left|\mathit{v}\mathbf{\times }\mathit{a}\right|\right|}{\left|\left|\mathit{v}\right|\right|}. \end{gathered}$$