Notes: A Rocket fired vertically burns its fuel for 25 seconds, which provides a thrust producing a constant upward acceleration of 8 m/s2. Determine (i)...

Saturday, May 28, 2016

A Rocket fired vertically burns its fuel for 25 seconds, which provides a thrust producing a constant upward acceleration of 8 m/s2. Determine (i)...

Denote the first 25 seconds as $\displaystyle{t}_{{1}}$ and the acceleration during this time as $\displaystyle{a}.$ Then the height is $\displaystyle{H}_{{1}}{\left({t}\right)}=\frac{{{a}{{t}}^{{2}}}}{{2}}$ and the speed is $\displaystyle{V}_{{1}}{\left({t}\right)}={a}{t}.$ So the final speed after fuel is burnt is $\displaystyle{v}_{{1}}={a}{t}_{{1}}$ and the final height is $\displaystyle{h}_{{1}}=\frac{{{a}{{t}_{{1}}^{{2}}}}}{{2}}.$

The height after $\displaystyle{t}_{{1}}$ is the usual free fall height:

$\displaystyle{H}_{{2}}{\left({t}\right)}={h}_{{1}}+{v}_{{1}}{\left({t}-{t}_{{1}}\right)}-\frac{{{{g{{\left({t}-{t}_{{1}}\right)}}}}^{{2}}}}{{2}}.$

The maximum altitude is reached at $\displaystyle{t}_{{2}}={t}_{{1}}+\frac{{v}_{{1}}}{{g{}}}$ and it is $\displaystyle{h}_{{2}}={h}_{{1}}+\frac{{{{v}_{{1}}^{{2}}}}}{{{2}{g}}}=\frac{{a}}{{2}}{{t}_{{1}}^{{2}}}{\left({1}+\frac{{a}}{{g}}\right)}.$

Rocket falls when $\displaystyle{H}_{{2}}{\left({t}\right)}={0},$ $\displaystyle{t}\gt{t}_{{1}}.$ This is a quadratic equation and the solution is

$\displaystyle{t}_{{3}}={t}_{{1}}+\frac{{{v}_{{1}}+\sqrt{{{{v}_{{1}}^{{2}}}+{4}\cdot\frac{{g}}{{2}}\cdot{h}_{{1}}}}}}{{{2}\cdot{\left(\frac{{g}}{{2}}\right)}}}={t}_{{1}}+\frac{{{v}_{{1}}+\sqrt{{{{v}_{{1}}^{{2}}}+{2}{g{{h}}}_{{1}}}}}}{{{g}}}=$

$\displaystyle={t}_{{1}}+\frac{{a}}{{{g}}}{t}_{{1}}{\left({1}+\sqrt{{{{\left(\frac{{a}}{{g}}\right)}}^{{2}}+\frac{{a}}{{g}}}}\right)}.$

The answers are: (i) $\displaystyle{h}_{{1}},$ (ii) $\displaystyle{h}_{{2}},$ (iii) $\displaystyle{t}_{{3}}.$ In numbers they are (i) 2500 m, (ii) about 4541 m and (iii) about 70 s.

Notes

Saturday, May 28, 2016

A Rocket fired vertically burns its fuel for 25 seconds, which provides a thrust producing a constant upward acceleration of 8 m/s2. Determine (i)...

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