Solution 8.6b

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Here we use
Here we use
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<math>\mathbf{r}=\mathbf{u}t+\frac{1}{2}\mathbf{a}{{t}^{\ 2}}+{{\mathbf{r}}_{0}} </math>
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<math>\mathbf{r}=\frac{1}{2}(\mathbf{u}+\mathbf{v})t+{{\mathbf{r}}_{0}}</math>
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 +
in the first part.
According to the text
According to the text
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<math>\mathbf{u}=\mathbf{i}+2\mathbf{j}</math>
<math>\mathbf{u}=\mathbf{i}+2\mathbf{j}</math>
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and part a) gave
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<math>\mathbf{v}=6\mathbf{i}-8\mathbf{j}</math>
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<math>\mathbf{a}=0\textrm{.}5\mathbf{i}-\mathbf{j}</math>
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We assume the starting point is the origin so that <math>{{\mathbf{r}}_{0}}=0</math>.
We assume the starting point is the origin so that <math>{{\mathbf{r}}_{0}}=0</math>.
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At <math>t=10+40=50</math> we obtain
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At <math>t=10</math>
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<math>\mathbf{r}=(\mathbf{i}+2\mathbf{j}) \times 50+\frac{1}{2}(\mathbf{a}=0\textrm{.}5\mathbf{i}-\mathbf{j} )\times{{50}^{\ 2}}</math>
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<math> \mathbf{r}=\frac{1}{2}(\mathbf{i}+2\mathbf{j}+6\mathbf{i}-8\mathbf{j})t=(3\textrm{.}5\mathbf{i}-3\mathbf{j}) \times 10=35\mathbf{i}-30\mathbf{j}=(7\mathbf{i}-6\mathbf{j}) \times 5</math>
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The distance is the magnitude of this vector
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<math>\sqrt{{{7}^{2}}+{{\left( -6 \right)}^{2}}}=\sqrt{13}=3\textrm{.}6\ \text{m}</math>
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The distance is the magnitude of this vector
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Thus during the first part the boat has travelled a distance 3.6 m.

Revision as of 19:54, 14 April 2010

Here we use

\displaystyle \mathbf{r}=\frac{1}{2}(\mathbf{u}+\mathbf{v})t+{{\mathbf{r}}_{0}}

in the first part.

According to the text

\displaystyle \mathbf{u}=\mathbf{i}+2\mathbf{j}

\displaystyle \mathbf{v}=6\mathbf{i}-8\mathbf{j}

We assume the starting point is the origin so that \displaystyle {{\mathbf{r}}_{0}}=0.

At \displaystyle t=10

\displaystyle \mathbf{r}=\frac{1}{2}(\mathbf{i}+2\mathbf{j}+6\mathbf{i}-8\mathbf{j})t=(3\textrm{.}5\mathbf{i}-3\mathbf{j}) \times 10=35\mathbf{i}-30\mathbf{j}=(7\mathbf{i}-6\mathbf{j}) \times 5

The distance is the magnitude of this vector

\displaystyle \sqrt{{{7}^{2}}+{{\left( -6 \right)}^{2}}}=\sqrt{13}=3\textrm{.}6\ \text{m}

Thus during the first part the boat has travelled a distance 3.6 m.

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