How will the momentum of a body change if we double its kinetic energy? Prove by using equations.

Friday, October 31, 2008

How will the momentum of a body change if we double its kinetic energy? Prove by using equations.

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For a body having the mass $\displaystyle{m}$ and the speed $\displaystyle{V}$ , its kinetic energy $\displaystyle{E}$ is equal to

$\displaystyle\frac{{{m}{{V}}^{{2}}}}{{2}},$

and its momentum (impulse) $\displaystyle{J}$ is equal to $\displaystyle{m}{V}.$

Doubling the kinetic energy of a body probably is achieved by changing its speed. If originally a body had the speed $\displaystyle{V}_{{0}},$ then its kinetic energy was $\displaystyle{E}_{{0}}=\frac{{{m}\cdot{{V}_{{0}}^{{2}}}}}{{2}},$

and its impulse was $\displaystyle{J}_{{0}}={m}\cdot{V}_{{0}}.$

And if we change its speed to $\displaystyle{V}_{{1}}$ in such a way that the energy doubles,

$\displaystyle{E}_{{1}}=\frac{{{m}\cdot{{V}_{{1}}^{{2}}}}}{{2}}={2}{E}_{{0}}={m}{{V}_{{0}}^{{2}}},$

then $\displaystyle{V}_{{1}}=\sqrt{{{2}}}\cdot{V}_{{0}}.$ And the new momentum will be

$\displaystyle{J}_{{1}}={m}\cdot{V}_{{1}}=\sqrt{{{2}}}\cdot{m}\cdot{V}_{{0}}=\sqrt{{{2}}}\cdot{J}_{{0}}.$

The answer: if the kinetic energy of a body doubles, then its impulse is multiplied by $\displaystyle\sqrt{{{2}}}.$ Note that $\displaystyle\sqrt{{{2}}}$ is approximately 1.4.

P.S. If we double the kinetic energy by changing the mass, then it must be doubled also and the momentum will be also doubled.

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Friday, October 31, 2008

How will the momentum of a body change if we double its kinetic energy? Prove by using equations.

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