A CHUVA E O PÁRA-BRISA

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚

❆ ❝❤✉✈❛ ❡ ♦ ♣❛r❛✲❜r✐s❛

❍♦s❡❛♥♦ ❈♦st❛ ❞❛ ❙✐❧✈❛

❘❡❧❛tór✐♦ ♣❛r❛ ♦ ❊①❛♠❡ ●❡r❛❧ ❞❡ ◗✉❛❧✐✜❝❛çã♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ▼❡✳ ▼ár✐♦ ●♦♠❡s ❞♦s ❙❛♥t♦s

(2)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❆ ❝❤✉✈❛ ❡ ♦ ♣❛r❛✲❜r✐s❛

❍♦s❡❛♥♦ ❈♦st❛ ❞❛ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ▼❡✳ ▼ár✐♦ ●♦♠❡s ❞♦s ❙❛♥t♦s

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❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❍♦s❡❛♥♦ ❈♦st❛ ❞❛ ❙✐❧✈❛

❆ ❝❤✉✈❛ ❡ ♦ ♣❛r❛✲❜r✐s❛

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡✲ r❛❧ ❞♦ P✐❛✉í ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ▼❡✳ ▼ár✐♦ ●♦♠❡s ❞♦s ❙❛♥t♦s ❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❊③❡q✉✐❛s ▼❛t♦s ❊st❡✈❡s ▼❛t❡♠át✐❝❛ ✲ ■❋P■

Pr♦❢✳❉r✳ ❏✉r❛♥❞✐r ❞❡ ❖❧✐✈❡✐r❛ ▲♦♣❡s ▼❛t❡♠át✐❝❛ ✲ ❯❋P■

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❊❯❙✱ ♣❡❧❛ ✈✐❞❛ ❡ ♣♦r t❡r ♠❡ ❣✉✐❛❞♦ ♣❡❧♦ ❝❛♠✐♥❤♦ ❝❡rt♦✳

➚ ▼■◆❍❆ ❋❆▼❮▲■❆✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♠ã❡ ✭▼❛r✐❛ ❏♦sé✮ ❡ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ✭❈❤❛r❧❡♥❡✮✳

❆❖❙ ❆▼■●❖❙ ❉❊ ▼❊❙❚❘❆❉❖ ▼✐❧t♦♥✱ ❈❤✐❝♦✱ ❊✈❛♥❞r♦ ❡ ❲②❧s♦♥ ♣♦✐s ❢♦r❛♠ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ♠❡✉ s✉❝❡ss♦ ♥♦ P❘❖❋▼❆❚✳

❆❖ ❈❖▲❊●❆ ❉❊ ❚❘❆❇❆▲❍❖ ❆♥tô♥✐♦ ❈❛r❧♦s✱ ♣♦✐s ♠❡ ❛❥✉❞♦✉ ♠✉✐t♦ ❝♦♠ ❞✐❝❛s ❡ s✉♣♦rt❡s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ s❡❣✉✐r ❡♠ ❢r❡♥t❡✳

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✏➱ ❜♦♠ t❡r ❞✐♥❤❡✐r♦ ❡ ❛s ❝♦✐s❛s q✉❡ ♦ ❞✐♥❤❡✐r♦ ♣♦❞❡ ❝♦♠♣r❛r✳ ▼❛s é ❜♦♠ t❛♠❜é♠ ✈❡r✐✜❝❛r ❞❡ ✈❡③ ❡♠ q✉❛♥❞♦ s❡ ♥ã♦ ❡st❛♠♦s ♣❡r❞❡♥❞♦ ❛s ❝♦✐s❛s q✉❡ ♦ ❞✐♥❤❡✐r♦ ♥ã♦ ♣♦❞❡ ❝♦♠♣r❛r✑

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❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ ♠♦str❛ ❝♦♠♦ ❞❡t❡r♠✐♥❛r ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛s ❣♦t❛s ❞❡ ❝❤✉✈❛✱ q✉❡ ❝❛❡♠ ❡♠ ✉♠ ♣❛r❛✲❜r✐s❛ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✱ ❢♦r♠❛♥❞♦ ✉♠ â♥❣✉❧♦ θ (0≤ θ ≤90◦₎ _❝♦♠

❛ ✈❡rt✐❝❛❧✳ ❆♣ós ❞❡t❡r♠✐♥❛r ❡ss❡ ❧✉❣❛r✱ s❡rá ❝❛❧❝✉❧❛❞♦ s❡✉ ✈♦❧✉♠❡✳ P❛r❛ ✐ss♦✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s tó♣✐❝♦s✱ q✉❡ s❡r✈✐rã♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛❀ t❛✐s ❝♦♠♦✿ ❞❡✜♥✐çã♦ ❡ ❡①❡♠♣❧♦s ❞❡ ❧✉❣❛r❡s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✱ ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r ❡ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛✳

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❆❜str❛❝t

❚❤✐s ♣❛♣❡r s❤♦✇s ❤♦✇ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❝✉s ♦❢ r❛✐♥❞r♦♣s✱ ❢❛❧❧✐♥❣ ♦♥ ❛ ✇✐♥❞s❤✐❡❧❞ ♦❢ ❛♥ ❛✉t♦♠♦❜✐❧❡✱ ❛t ❛♥ ❛♥❣❧❡ ♦❢ θ ✭0≤ θ ≤90◦_{✮ ❢r♦♠ t❤❡ ✈❡rt✐❝❛❧✳ ❆❢t❡r ❞❡t❡r♠✐♥✐♥❣}

t❤✐s ♣❧❛❝❡✱ ✐ts ✈♦❧✉♠❡ ✇✐❧❧ ❜❡ ❝❛❧❝✉❧❛t❡❞✳ ❋♦r t❤✐s✱ s♦♠❡ t♦♣✐❝s t❤❛t ✇✐❧❧ s❡r✈❡ ❛s t♦♦❧s ❢♦r ♣r♦❜❧❡♠ s♦❧✈✐♥❣ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞❀ s✉❝❤ ❛s t❤❡ ❞❡✜♥✐t✐♦♥ ❛♥❞ ❡①❛♠♣❧❡s ♦❢ ❧♦❝✐✱ s✉❝❤ ❛s ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ ♣r✐s♠ ✈❡❝t♦r ❝♦♠♣♦♥❡♥ts ❛♥❞ r❡❧❛t✐✈❡ s♣❡❡❞✳

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❈❤✉✈❛ ❢❛③❡♥❞♦ ✉♠ â♥❣✉❧♦θ ❝♦♠ ❛ ✈❡rt✐❝❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶ ❉✐s❝♦ ❞❡ ❝❡♥tr♦A ❡ r❛✐♦ r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❖ ♣❧❛♥♦ ♠❡❞✐❛❞♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❚r❛♣é③✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❚r✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ Pr✐s♠❛ ❖❜❧íq✉♦ ❞❡ ❜❛s❡ P ❡ ❛❧t✉r❛ ❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ ❙ó❧✐❞♦s ❝♦rt❛❞♦s ♣♦r ✉♠ ♣❧❛♥♦ ❤♦r✐③♦♥t❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✼ ❱❡t♦r~v =B−A✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✽ ❱❡t♦r❡s ♦♣♦st♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✾ ❱❡t♦r❡s ♣❛r❛❧❡❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✵ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✶ ❈♦r♣♦s ❡♠ ♠♦✈✐♠❡♥t♦s ❞❡ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦s ♦♣♦st♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶ ❙✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ■♥✜♥✐❞❛❞❡s ❞❡ ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠O✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ❙❡❣♠❡♥t♦OO,_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶}

✸✳✹ ▲✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛s ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠B✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ❱✐st❛ ❧❛t❡r❛❧ ❞♦ ♣r✐s♠❛ ❞❡ ❜❛s❡ B ❡ ❛❧t✉r❛ h✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✻ ❉❡s❧♦❝❛♠❡♥t♦ h ❞❡ ✉♠ ♠ó✈❡❧✱ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ Vrn ❡♠ T ✉♥✐❞❛❞❡s ❞❡

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✾

✶✳✶ ❖ ♣r♦❜❧❡♠❛ ♣❛r❛ ✉♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷ ▼♦❞❡❧❛❣❡♠ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✷ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✷✶

✷✳✶ ▲✉❣❛r ●❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❚r✐❣♦♥♦♠❡tr✐❛ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ Pr✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹✳✶ ❱♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺ ❱❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺✳✶ ❱❡t♦r ♦♣♦st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺✳✷ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺✳✸ P❛r❛❧❡❧✐s♠♦ ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺✳✹ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r ❡♠ r❡❧❛çã♦ ❛♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s

❝❛rt❡s✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✺✳✺ ❱❡❧♦❝✐❞❛❞❡ ❘❡❧❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸ ❆ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✷✾

✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✸✼

❘❡❢❡rê♥❝✐❛s ✸✾

(19)

(20)

✶ ■♥tr♦❞✉çã♦

◆❛ r❡✈✐st❛ ❞♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ ♥♦ _{✶✽ ✭ ❘P▼ ✮✱ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♦ ❛rt✐❣♦ ❛}

❝❤✉✈❛ ❡ ♦ ♣❛r❛✲❜r✐s❛✱ ❝✉❥♦ ❛✉t♦r é ◆❡② ❈❛r♠♦♥❛ ❥✉♥✐♦r✳ ❖ ❛rt✐❣♦ ♠♦str❛ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ❝❤✉✈❛ q✉❡ ✐♥❝✐❞❡ ✈❡rt✐❝❛❧♠❡♥t❡ ♥♦ ♣❛r❛✲❜r✐s❛ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✳ ❆ s♦❧✉çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ ❢♦✐ ❢❡✐t❛ ✉s❛♥❞♦ ❢❡rr❛♠❡♥t❛s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ t❛✐s ❝♦♠♦✿ ●❡♦♠❡tr✐❛✱ ❚r✐❣♦♥♦♠❡tr✐❛✱ ❋✉♥çõ❡s✱ ❱❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛✳

◆♦ ✜♥❛❧ ❞♦ ❛rt✐❣♦ ♦ ❛✉t♦r s✉❣❡r✐✉ q✉❡ ❢♦ss❡ ❢❡✐t♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡❧❡❀ ✐st♦ é✱ s♦❧✉❝✐♦♥❛ss❡ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ✉♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧✳ ❊♠ s❡ tr❛t❛♥❞♦ ❞❡ ✉♠ ♣r♦✲ ❜❧❡♠❛ ❝✉❥♦ r❡s✉❧t❛❞♦ é ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ s✐♠♣❧❡s✱ ♠❛s q✉❡ ❞❡s♣❡rt❛ ❝✉r✐♦s✐❞❛❞❡✱ r❡s♦❧✈✐ tr❛❜❛❧❤❛r ♦ ♣r♦❜❧❡♠❛✳

P❛r❛ ❡♥t❡♥❞❡r ♦ q✉❡ é ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ s❡❣✉❡✲s❡ ❛ ❞❡✜♥✐çã♦✿ ▼♦❞❡❧♦ ▼❛t❡✲ ♠át✐❝♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ sí♠❜♦❧♦s ❡ r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ r❡♣r❡s❡♥t❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♦ ♦❜❥❡t♦ ❡st✉❞❛❞♦ ❬✶❪✳

✶✳✶ ❖ ♣r♦❜❧❡♠❛ ♣❛r❛ ✉♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧

❖ ♣r♦❜❧❡♠❛ ❛❣♦r❛✱ ❝♦♥s✐st❡ ❡♠ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ t♦t❛❧ ❞❡ ❝❤✉✈❛ q✉❡ ✐♥❝✐❞❡✱ ❢♦r✲ ♠❛♥❞♦ ✉♠ â♥❣✉❧♦ θ ❝♦♠ ❛ ✈❡rt✐❝❛❧✱ ♥♦ ♣❛r❛✲❜r✐s❛ ❞❡ ✉♠ ❝❛rr♦ ❞✉r❛♥t❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❝♦♥s✐❞❡r❛❞♦❀ s❡♥❞♦ q✉❡ ❛ ❝❤✉✈❛ ❡ ♦ ❝❛rr♦ ♣♦ss✉❡♠ ✈❡❧♦❝✐❞❛❞❡s ❝♦♥st❛♥t❡s✳

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✷✵ ■♥tr♦❞✉çã♦

✶✳✷ ▼♦❞❡❧❛❣❡♠ ❞♦ ♣r♦❜❧❡♠❛

P❛r❛ ♠♦❞❡❧❛❣❡♠ ❞❛ s✐t✉❛çã♦✱ r❡♣r❡s❡♥t❛♠♦s ♦ ♣❛r❛✲❜r✐s❛ ♣♦r ✉♠❛ ár❡❛ ♣❧❛♥❛ ❡ ❢❡❝❤❛❞❛ ❇✱ ❝✉❥♦ ♣❧❛♥♦ ❢♦r♠❛ ✉♠ â♥❣✉❧♦ α ❝♦♠ ❛ ❤♦r✐③♦♥t❛❧ ❡ q✉❡ s❡ ❞❡s❧♦❝❛ ❤♦r✐③♦♥✲ t❛❧♠❡♥t❡ ❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ Vp ♣♦r ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❝♦♠ ❛♠♣❧✐t✉❞❡

❚✳ ❊♥q✉❛♥t♦ ✐st♦✱ ♥❛ r❡❣✐ã♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❡ ♥♦ ✐♥t❡r✈❛❧♦ ❝♦♥s✐❞❡r❛❞♦✱ ♦❝♦rr❡ ✉♠❛ ❝❤✉✈❛ ❝♦♥tí♥✉❛ q✉❡ s❡ ♣r❡❝✐♣✐t❛ ❢♦r♠❛♥❞♦ ✉♠ â♥❣✉❧♦ θ✱ ❝♦♠ θ ✈❛r✐❛♥❞♦ ♥♦ ✐♥t❡r✈❛❧♦

0≤θ≤90◦ ❝♦♠ ❛ ✈❡rt✐❝❛❧ ❛ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ Vg✳ ❈♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✳

❋✐❣✉r❛ ✶✳✶✿ ❈❤✉✈❛ ❢❛③❡♥❞♦ ✉♠ â♥❣✉❧♦ θ ❝♦♠ ❛ ✈❡rt✐❝❛❧✳

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝❛❧❝✉❧❛r✱ ♥✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ T✱ ♦ ✈♦❧✉♠❡ ❞❡ ❝❤✉✈❛ q✉❡ ❝❛✐ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ Vg ❢♦♠❛♥❞♦ ✉♠ â♥❣✉❧♦ θ ❝♦♠ ❛ ✈❡rt✐❝❛❧✱ ❡♠

✉♠ ♣❛r❛✲❜r✐s❛ ❞❡ ✉♠ ❝❛rr♦✱ q✉❡ ❛♥❞❛ ❝♦♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ Vp✳ P❛r❛ ✐ss♦✱

(22)

✷ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

✷✳✶ ▲✉❣❛r ●❡♦♠étr✐❝♦

❉❡✜♥✐çã♦ ✷✳✶✳ ▲✉❣❛r ●❡♦♠étr✐❝♦ é ♦ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡ ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ♦✉ ❞♦ ❡s♣❛ç♦ q✉❡ ❣♦③❛♠ ❞❡ ✉♠❛ ♠❡s♠❛ ♣r♦♣r✐❡❞❛❞❡✳

❊①❡♠♣❧♦ ✷✳✶✳ ❖ ❞✐s❝♦ ❉ ❞❡ ❝❡♥tr♦A= (a, b)❡ r❛✐♦ré ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦sP = (x, y)

❝✉❥❛ ❞✐stâ♥❝✐❛ ❛♦ ♣♦♥t♦ ❆ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ r✳ ■st♦ é✿

D={(x, y)∈R2/(x−a)2_{+ (}_y₋_b₎2 _≤_r2_}

❋✐❣✉r❛ ✷✳✶✿ ❉✐s❝♦ ❞❡ ❝❡♥tr♦ A ❡ r❛✐♦ r✳

❊①❡♠♣❧♦ ✷✳✷✳ P❧❛♥♦ ♠❡❞✐❛❞♦r✳ ➱ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ q✉❡ sã♦ ❡q✉✐❞✐st❛♥t❡s ❞❡ ❞♦✐s ♣♦♥t♦s A ❡ B✳

(23)

✷✷ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

❋✐❣✉r❛ ✷✳✷✿ ❖ ♣❧❛♥♦ ♠❡❞✐❛❞♦r✳

✷✳✷ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s

❚❡♦r❡♠❛ ✷✳✶✳ ◆✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❞❡ ❝❛t❡t♦s a✱ b❀ ♦ q✉❛❞r❛❞♦ ❞❛ ❤✐♣♦t❡♥✉s❛c é ✐❣✉❛❧ ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s ❝❛t❡t♦s✳

P❛r❛ ❞❡♠♦♥str❛çã♦ ❝♦♥s✐❞❡r❡ ❛ ✜❣✉r❛ ✷✳✸✳

❋✐❣✉r❛ ✷✳✸✿ ❚r❛♣é③✐♦✳

❉❡♠♦♥str❛çã♦✳ ❆ àr❡❛ ❞♦ tr❛♣é③✐♦ ❝♦♠ ❜❛s❡s a✱ b✱ ❡ ❛❧t✉r❛ a+b é ✐❣✉❛❧ ❛ ♠❡t❛❞❡ ❞❛ s♦♠❛ ❞❛s ❜❛s❡s ✈❡③❡s ❛ ❛❧t✉r❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ♠❡s♠❛ ár❡❛ é t❛♠❜é♠ ✐❣✉❛❧ à s♦♠❛ ❞❛s ár❡❛s ❞♦s três tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✳ ■st♦ é✿

a+b

2 .(a+b) =

(24)

❚r✐❣♦♥♦♠❡tr✐❛ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✷✸

❞❡s❡♥✈♦❧✈❡♥❞♦ ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ♦❜t❡♠♦s a2 2 + ab 2 + ab 2 + b2 2 = ab 2 + ab 2 + c2 2 ⇒ a2 2 + b2 2 = c2

2 ⇒c

2 ₌_a2₊_b2

❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

✷✳✸ ❚r✐❣♦♥♦♠❡tr✐❛ ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦

❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ABC✱ r❡tâ♥❣✉❧♦ ❡♠ A✿

C A B a c b

❋✐❣✉r❛ ✷✳✹✿ ❚r✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✳

❖ s❡❣♠❡♥t♦BC ❞❡ ♠❡❞✐❞❛a✱ ♦♣♦st♦ ❛♦ â♥❣✉❧♦ r❡t♦A✱ é ❝❤❛♠❛❞♦ ❞❡ ❤✐♣♦t❡♥✉s❛❀b ♦s s❡❣♠❡♥t♦s CA ❡AB✱ ❞❡ ♠❡❞✐❞❛s b ❡ c✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝❛t❡t♦s✳ ◆❡ss❡ tr✐â♥❣✉❧♦✱ ❡stã♦ ❞❡✜♥✐❞❛s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ♣❛r❛ ♦ â♥❣✉❧♦ ❛❣✉❞♦ _B✳b

• senBb= b a

• cosBb = c a

• tgBb = b_c

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❡♥❝♦♥tr❛✲s❡ ❛s r❡❧❛çõ❡s ♣❛r❛ ♦ â♥❣✉❧♦ ❛❣✉❞♦C✳b

✷✳✹ Pr✐s♠❛

❈♦♥s✐❞❡r❡ ✉♠ ♣♦❧í❣♦♥♦ P✱ ❝♦♥✈❡①♦✱ ❝♦♥t✐❞♦ ❡♠ ✉♠ ♣❧❛♥♦ α ❡ ✉♠ s❡❣✉✐♠❡♥t♦ AB ❝♦♠ ❛♣❡♥❛s ✉♠❛ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s ♣❡rt❡♥❝❡♥t❡ ❛ α✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✺✳ ❉❡✜♥✐çã♦ ✷✳✷✳ ❈❤❛♠❛✲s❡ ♣r✐s♠❛ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s s❡❣✉✐♠❡♥t♦s ♣❛r❛❧❡❧♦s ❡ ❝♦♥❣r✉✲ ❡♥t❡s ❛ AB✱ ❝♦♠ ✉♠❛ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s ❡♠ P ❡ ❝♦♥t✐❞♦ ♥♦ ♠❡s♠♦ s❡♠✐❡s♣❛ç♦ ❡♠ q✉❡ ❡stá ❝♦♥t✐❞♦ ♦ s❡❣✉✐♠❡♥t♦ AB✳

❆①✐♦♠❛✭Pr✐♥❝í♣✐♦ ❞❡ ❝❛✈❛❧✐❡r✐✮

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✷✹ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

❋✐❣✉r❛ ✷✳✺✿ Pr✐s♠❛ ❖❜❧íq✉♦ ❞❡ ❜❛s❡ P ❡ ❛❧t✉r❛ ❤✳

✷✳✹✳✶ ❱♦❧✉♠❡

❈♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♣♦❞❡♠♦s ♦❜t❡r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✳ ❈♦♥s✐❞❡r❡ ✉♠ ♣r✐s♠❛ ❞❡ ❛❧t✉r❛ h✱ ❝✉❥❛ ❜❛s❡ s❡❥❛ ✉♠ ♣♦❧í❣♦♥♦ P ❞❡ ár❡❛ A ❝♦♥t✐❞♦ ♥✉♠ ♣❧❛♥♦ ❤♦r✐③♦♥t❛❧✳ ❈♦♥str✉í♠♦s ❛♦ ❧❛❞♦ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦ ❝♦♠ ♠❡s♠❛ ❛❧t✉r❛ h ❡ ❞❡ ❢♦r♠❛ q✉❡ s✉❛ ❜❛s❡ s❡❥❛ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛A✳ ❆❣♦r❛ s✉♣♦♥❤❛ q✉❡ ♦s ❞♦✐s só❧✐❞♦s s❡❥❛♠ ❝♦rt❛❞♦s ♣♦r ♦✉tr♦ ♣❧❛♥♦ ❤♦r✐③♦♥t❛❧✱ q✉❡ ♣r♦❞✉③ s❡❝çõ❡s ❞❡ ár❡❛s A1 ❡ A2 ♥♦

♣r✐s♠❛ ❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▼❛s ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ é t❛♠❜é♠ ✉♠ ♣r✐s♠❛ ❡✱ s❛❜❡♠♦s q✉❡ ❡♠ t♦❞♦ ♣r✐s♠❛✱ ✉♠❛ s❡❝çã♦ ♣❛r❛❧❡❧❛ à ❜❛s❡ é ❝♦♥❣r✉❡♥t❡ ❝♦♠ ❡ss❛ ❜❛s❡✳ ▲♦❣♦✱ ❝♦♠♦ ✜❣✉r❛s ❝♦♥❣r✉❡♥t❡s tê♠ ♠❡s♠❛ ár❡❛✱ t❡♠♦s q✉❡ A1 = A = A2 ❡✱

♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝❛✈❛❧✐❡r✐✱ ♦s ❞♦✐s só❧✐❞♦s tê♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❈♦♠♦ ♦ ✈♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ éA.h✱ ♦ ✈♦❧✉♠❡ ❞♦ ♣r✐s♠❛ é t❛♠❜é♠ ♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❡ s✉❛ ❜❛s❡ ♣♦r s✉❛ ❛❧t✉r❛✳

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❱❡t♦r ✷✺

✷✳✺ ❱❡t♦r

❉❡✜♥✐çã♦ ✷✳✸✳ ❱❡t♦r é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡❣♠❡♥t♦s ♦r✐❡♥t❛❞♦s ❞❡ ♠❡s♠❛ ❞✐r❡çã♦✱ ❞❡ ♠❡s♠♦ s❡♥t✐❞♦ ❡ ❞❡ ♠❡s♠♦ ♠ó❞✉❧♦✳

❆ ✜❣✉r❛ ✷✳✼ ❡stá r❡♣r❡s❡♥t❛❞♦ ✉♠ ✈❡t♦r~v✳

❋✐❣✉r❛ ✷✳✼✿ ❱❡t♦r~v =B−A✳

✷✳✺✳✶ ❱❡t♦r ♦♣♦st♦

❉❡✜♥✐çã♦ ✷✳✹✳ ❉❛❞♦ ✉♠ ✈❡t♦r −→AB ♦ s❡✉ ♦♣♦st♦ é ♦ ✈❡t♦r −→BA ❡ s❡ ✐♥❞✐❝❛ ♣♦r −−→AB✳

❖ ✈❡t♦r ♦♣♦st♦ ❞❡ ✉♠ ✈❡t♦r ~v é r❡♣r❡s❡♥t❛❞♦ ♣♦r −~v✳

❋✐❣✉r❛ ✷✳✽✿ ❱❡t♦r❡s ♦♣♦st♦s✳

✷✳✺✳✷ ▼ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r

❉❡✜♥✐çã♦ ✷✳✺✳ ➱ ♦ ♥ú♠❡r♦ ♥ã♦ ♥❡❣❛t✐✈♦ q✉❡ ✐♥❞✐❝❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r✳ ❖ ♠ó❞✉❧♦ ❞❡ ✉♠ ✈❡t♦r~u é ✐♥❞✐❝❛❞♦ ♣♦r ||~u||.

✷✳✺✳✸ P❛r❛❧❡❧✐s♠♦ ❞❡ ✈❡t♦r❡s

❉❡✜♥✐çã♦ ✷✳✻✳ ❉♦✐s ✈❡t♦r❡s ~u ❡ ~v sã♦ ❞✐t♦s ♣❛r❛❧❡❧♦s q✉❛♥❞♦ ♣♦ss✉✐r❡♠ ❛ ♠❡s♠❛ ❞✐r❡çã♦✳

(27)

✷✻ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

❋✐❣✉r❛ ✷✳✾✿ ❱❡t♦r❡s ♣❛r❛❧❡❧♦s✳

✷✳✺✳✹ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r ❡♠ r❡❧❛çã♦ ❛♦ s✐st❡♠❛ ❞❡ ❝♦✲

♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s

❙❡❥❛ ~u ✉♠ ✈❡t♦r q✉❛❧q✉❡r✳ ❚♦♠❛♠♦s ❛ ♦r✐❣❡♠ ❞❡ ~u ♥♦ ♣♦♥t♦ 0✳ ❈❤❛♠❛✲s❡ ❝♦♠✲

♣♦♥❡♥t❡s ❞❡ ~u s❡❣✉♥❞♦ ♦s ❡✐①♦s 0x ❡ 0y ❛s ♣r♦❥❡çõ❡s ux ❡ uy ❞❡ ~u s♦❜r❡ ❡ss❡s ❡✐①♦s✳

❋✐❣✉r❛ ✷✳✶✵✿ ❈♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r✳

• ❖ ♠ó❞✉❧♦ ♦✉ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ~u é ❞❛❞♦ ♣♦r✿

||~u||=qu2

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❱❡t♦r ✷✼

• ❙❡♥❞♦ α ♦ â♥❣✉❧♦ ❡♥tr❡ ~u ❡ Ox✱ t❡♠ s❡

(

ux =||~u||.cosα

uy =||~u||.senα

✷✳✺✳✺ ❱❡❧♦❝✐❞❛❞❡ ❘❡❧❛t✐✈❛

❉❡✜♥✐çã♦ ✷✳✼✳ ❖ ♠ó❞✉❧♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❞❡ ✉♠ ❝♦r♣♦ A❡♠ r❡❧❛çã♦ ❛♦ ❝♦r♣♦ B q✉❡ s❡ ♠♦✈❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s VA ❡ VB ❞❡ ♠❡s♠❛ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦s

♦♣♦st♦s é ❞❛❞♦ ♣♦r ||VAB||=||VA||+||VB||

❊①❡♠♣❧♦ ✷✳✸✳ ◆❛ ✜❣✉r❛ ❛❝✐♠❛ ✷✳✶✶✳ ❙❡ ♦s ❝♦s♣♦s ❆ ❡ ❇ s❡ ❞❡s❧♦❝❛♠ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s ❞❡ ♠ó❞✉❧♦ ||VA||= 10m/s❡ ||VB||= 5m/s✱ ❡♥tã♦ ❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❞♦ ❝♦r♣♦ ❆ ❡♠

r❡❧❛çã♦ ❛♦ ❝♦r♣♦ ❇ é ❞❛❞❛ ♣♦r✿ ||VAB||= 10 + 5 = 15m/s.

(29)

(30)

✸ ❆ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦✱ ❛ r❡s♦❧✉çã♦ s❡rá ❞✐✈✐❞✐❞❛❞❛ ❡♠ ❞✉❛s ♣❛rt❡s✿ Pr✐♠❡✐r❛ ♣❛rt❡

◆❡ss❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ s❡rá ❞❡t❡r♠✐♥❛❞♦ ❡♠ t = 0 ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❡ t♦❞❛s ❛s

❣♦t❛s ❞❡ ❝❤✉✈❛ q✉❡ t♦❝❛rã♦ ❛ s✉♣❡r❢í❝✐❡ B✱ ❡♠ ❛❧❣✉♠ ✐♥st❛♥t❡ t ❞♦ ✐♥t❡r✈❛❧♦ (0, T]✳

P❛r❛ ✐ss♦✱ t♦♠❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❝♦♠ ♦r✐❣❡♠ O ✜①❛ ❡ q✉❛❧q✉❡r✱ ♥❛ s✉♣❡r❢í❝✐❡ ❇ ❝♦❡r❡♥t❡ ❝♦♠ ❛s ❞✐r❡çõ❡s ❤♦r✐③♦♥t❛❧ ❡ ✈❡rt✐❝❛❧ ❞♦ ❧✉❣❛r✳ ❈♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✸✳✶✳

❋✐❣✉r❛ ✸✳✶✿ ❙✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳

P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ r❡s♦❧✉çã♦ ✐♥✐❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛✱ ✈❛♠♦s t❡r ❝♦♠♦ r❡❢❡rê♥❝✐❛ ✈✐s✉❛❧ ❛ ✜❣✉r❛ ✸✳✷✳

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✸✵ ❆ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

❋✐❣✉r❛ ✸✳✷✿ ■♥✜♥✐❞❛❞❡s ❞❡ ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠O✳

• α é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥çã♦ ❞♦ ♣❛r❛✲❜r✐s❛ ❝♦♠ ❛ ❤♦r✐③♦♥t❛❧❀

• β é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥❛çã♦ ❞❛ ❞✐stâ♥❝✐❛d ❝♦♠ ♦ ❡✐①♦ x❀

• θ é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥❛çã♦ ❞❛ ❝❤✉✈❛ ❝♦♠ ❛ ✈❡rt✐❝❛❧❀

• Vg é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❝❤✉✈❛❀

• Vp é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ O❀

• Vv é ❛ ❝♦♠♣♦♥❡♥t❡ ♥❛ ❞✐r❡çã♦ ✈❡rt✐❝❛❧ ❞❛ ✈❡❧♦❝✐❞❛❞❡ Vg❀

• Vh é ❛ ❝♦♠♣♦♥❡♥t❡ ♥❛ ❞✐r❡çã♦ ❤♦r✐③♦♥t❛❧ ❞❛ ✈❡❧♦❝✐❞❛❞❡ Vg✳

❖ ♣♦♥t♦ O s❡rá ♣♦♥t♦ ❢✉t✉r♦ ❞❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❣♦t❛s ❞♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❡ ✉♠❛ ❣♦t❛ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥♦ ♣♦♥t♦ G = (x, y) ∈ XOY ❡st❡❥❛ ♥♦ ♣♦♥t♦ O ∈XOY ❛♣ós ✉♠ t❡♠♣♦ t é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ ♦ t❡♠♣♦ tg ❣❛st♦ ♣❡❧❛

❣♦t❛ ♣❛r❛ ♣❡r❝♦rr❡r ✈❡rt✐❝❛❧♠❡♥t❡ ❛ ❞✐stâ♥❝✐❛ys❡❥❛ ✐❣✉❛❧ ❛♦ t❡♠♣♦ tp ❣❛st♦ ♣❡❧♦ ♣♦♥t♦

O ♣❛r❛ ♣❡r❝♦rr❡r ❤♦r✐③♦♥t❛❧♠❡♥t❡ ❛ ❞✐stâ♥❝✐❛x❝♦♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❤♦r✐③♦♥t❛❧ VH ❡♠ r❡❧❛çã♦ ❛ ❣♦t❛ ●✱ ❈♦♠ 0< tg =tp ≤T✳ ■st♦ é❀

tg =tp

P❡❧❛ ❞❡✜♥✐çã♦ ✭ ✷✳✼ ✮ ❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❤♦r✐③♦♥t❛❧ ❞♦ ♣♦♥t♦O ❡♠ r❡❧❛çã♦ ❛ ❣♦t❛ ● é ❞❛❞❛ ♣♦r

||VH||=||Vp||+||Vh||

❡ s❡

||Vv||=

y tg ❡

||VH||=

(32)

✸✶

❡♥tã♦ y

||Vv|| =

x

||VH||✳ P♦rt❛♥t♦

y =

||Vv||

||VH||

x. ✭✸✳✶✮

▼❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡Vg ♥❛s ❞✐r❡çõ❡s ✈❡rt✐❝❛❧ ❡ ❤♦r✐③♦♥t❛❧✱ sã♦ ❞❛❞❛s r❡s♣❡❝t✐✈❛✲

♠❡♥t❡ ♣♦r✿

||Vv||=||Vg||.cosθ ❡ ||Vh||=||Vg||.senθ. ✭✸✳✷✮

❉❛s ❡q✉❛çõ❡s ✭✸✳✶✮ ❡ ✭✸✳✷✮✱ r❡s✉❧t❛ ❡♠ y=

||Vg||.cosθ

||Vp||+||Vg||.senθ

x. ✭✸✳✸✮

❆ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❛♥❛❧ít✐❝❛ ❞❛ s❡♠✐✲r❡t❛ ❞❡ ♦r✐❣❡♠ O✳ ❆ ♣❛rt✐r ❞❡❧❛ ❞❡❞✉③✲s❡ q✉❡✱ ❛s ❣♦t❛s G = (x, y) ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ O✱ ❡♠ ❛❧❣✉♠ ✐♥st❛♥t❡ t ∈(0, T]✱ s❡ s✐t✉❛♠✱ ❡♠t = 0✱ s♦❜r❡ ♦ s❡❣♠❡♥t♦ OO,_{❞❛ s❡♠✐ r❡t❛ ❞❡ ♦r✐❣❡♠} _O _{❡ t❛①❛}

❞❡ ✈❛r✐❛çã♦ ||Vg||cosθ

||Vp||+||Vg||.senθ

✳

❆❣♦r❛ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ d✳ P❛r❛ ✐ss♦ ✈❡❥❛ ❛ ✜❣✉r❛ ✸✳✸✳

❋✐❣✉r❛ ✸✳✸✿ ❙❡❣♠❡♥t♦ OO,_✳

❖ ❝♦♠♣r✐♠❡♥t♦d ❞♦ s❡❣✉✐♠❡♥t♦OO, _{♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ t❡♦r❡♠❛}

❞❡ P✐tá❣♦r❛s ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ❣♦t❛ ♥❛ ♣♦s✐çã♦ O, _{é ❣♦t❛ q✉❡ ❛t✐♥❣❡ ♦ ♣♦♥t♦} _O _❞❛

s✉♣❡r❢í❝✐❡ ❇ ❛♣ós ✉♠ t❡♠♣♦ t=T✳ P♦r P✐tá❣♦r❛s t❡♠♦s q✉❡✿ d=px2₊_y2 ❡ ❝♦♠♦ _x_{= (}_||_V

p||+||Vg||.senθ).T ❡ y=||Vg||cosθ.T

s❡❣✉❡✲s❡ q✉❡

d =

q

[(||Vp||+||Vg||.senθ).T]2+ (||Vg||cosθ.T)2;

q✉❡ ❞❡s❡♥✈♦❧✈✐❞❛ r❡s✉❧t❛ ❡♠

(33)

✸✷ ❆ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

❈♦♠♦ ♦ ♣♦♥t♦ ❖ ❞❡ ❇ é q✉❛❧q✉❡r s❡❣✉❡✲s❡ q✉❡ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛s ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠ ❇ s❡ ❝♦♥st✐t✉✐ ❡♠ ✉♠ ♣r✐s♠❛ ♦❜❧íq✉♦ ❞❡ ❜❛s❡ ❇ ❡ ❛r❡st❛s ❧❛t❡r❛✐s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞ ❡ ✐♥❝❧✐♥❛çã♦

β =arctg

||Vg||

||Vp||+||Vg||.senθ

cosθ

; ✭✸✳✺✮

❡♠ r❡❧❛çã♦ ❛ ❤♦r✐③♦♥t❛❧✳ P❛r❛ ✈✐s✉❛❧✐③❛r ❡ss❡ ♣r✐s♠❛✱ ✈❡❥❛♠ ❛ ✜❣✉r❛ ✸✳✹✳

❋✐❣✉r❛ ✸✳✹✿ ▲✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛s ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠B✳

❋❛③❡♥❞♦ θ = 0 ♥❛s ❡q✉❛çõ❡s ✭ ✸✳✹ ✮ ❡ ✭ ✸✳✺ ✮ ❝❛✐ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❛rt✐❣♦ q✉❡

♠♦t✐✈♦✉ ❡ss❛ ❞✐ss❡rt❛çã♦✳ ❙❡❣✉♥❞❛ ♣❛rt❡

❈á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦✳

❉❡ ✉♠ ❧❛❞♦✱ s❡ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛s ❣♦t❛s ❝♦♠ ♣♦♥t♦ ❢✉t✉r♦ ❡♠ B é ✉♠ ♣r✐s♠❛ ❞❡ ❜❛s❡ ❇ ❡ ❛r❡st❛ ❧❛t❡r❛❧ ❞✱ ❡♥tã♦ s❡✉ ✈♦❧✉♠❡ s❡rá ❞❛❞♦ ♣♦r✿

V =A(B).h; ✭✸✳✻✮

(34)

✸✸

❖ ♣r♦❜❧❡♠❛ ❛❣♦r❛ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r s✉❛ ❛❧t✉t❛h✱ ♣❛r❛ ✐ss♦ ♦❜s❡r✈❡♠♦s ❛ ✜❣✉r❛

3.5✳

❋✐❣✉r❛ ✸✳✺✿ ❱✐st❛ ❧❛t❡r❛❧ ❞♦ ♣r✐s♠❛ ❞❡ ❜❛s❡ B ❡ ❛❧t✉r❛ h✳

❊❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ❛ ✜❣✉r❛ ✸✳✺✳

• α é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥çã♦ ❞♦ ♣❛r❛✲❜r✐s❛ ❝♦♠ ❛ ❤♦r✐③♦♥t❛❧❀

• B é ❛ ❜❛s❡ ❞♦ ♣r✐s♠❛❀

• θ é ♦ â♥❣✉❧♦ ❞❡ ✐♥❝❧✐♥❛çã♦ ❞❛ ❝❤✉✈❛ ❝♦♠ ❛ ✈❡rt✐❝❛❧❀

• Vg é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❝❤✉✈❛❀

• Vp é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ O❀

• Vgn é ♦ ♠ó❞✉❧♦ ❞❛ ❝♦♠♣♦♥❡♥t❡ ♥♦r♠❛❧ ❛♦ ♣❧❛♥♦ ❞❡ ❇ ❞❛ ✈❡❧♦❝✐❞❛❞❡Vg❀

• Vpn é ♦ ♠ó❞✉❧♦ ❞❛ ❝♦♠♣♦♥❡♥t❡ ♥♦r♠❛❧ ❛♦ ♣❧❛♥♦ ❞❡ ❇ ❞❛ ✈❡❧♦❝✐❞❛❞❡Vp❀

• h é ❛ ❛❧t✉r❛ ❞♦ ♣r✐s♠❛✳

◆❛ ✜❣✉r❛ ✸✳✺ s❛❥❛ M ♦ ♣♦♥t♦ ❞❡ B q✉❡ ❡stá ♠❛✐s ❞✐st❛♥t❡ ❞♦ ♣❧❛♥♦ ❤♦r✐③♦♥t❛❧✱ ❡ s❡❥❛ M, _{❛ ❣♦t❛ ❞❡ ❝❤✉✈❛ q✉❡ ❡♠} _t ₌_T _{❡st❛rá s♦❜r❡}_M _❡♠ _B_✳

❙❡♥❞♦ Vr ❛ ✈❡❧♦❝✐❞❛❞❡ r❡❧❛t✐✈❛ ❞❡ M ❡♠ r❡❧❛çã♦ ❛ M, ❡ Vrn ❛ s✉❛ ❝♦♠♣♦♥❡♥t❡ ♥❛

❞✐r❡çã♦ ♥♦r♠❛❧ ❛ B✱ ❡♥tã♦ ❛ ❛❧t✉r❛ h ❞♦ ♣r✐s♠❛ é ❛ ❞✐stâ♥❝✐❛ q✉❡ ✉♠ ♠ó✈❡❧ ♣❡r❝♦rr❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ Vrn ❡♠ T ✉♥✐❞❛❞❡s ❞❡ t❡♠♣♦✳

(35)

✸✹ ❆ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

❋✐❣✉r❛ ✸✳✻✿ ❉❡s❧♦❝❛♠❡♥t♦h❞❡ ✉♠ ♠ó✈❡❧✱ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡Vrn❡♠T ✉♥✐❞❛❞❡s ❞❡ t❡♠♣♦✳

❆ ❛❧t✉r❛ h é ✉♠ ❝❛t❡t♦ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉r♦ q✉❡ t❡♠ ❝♦♠♦ ❤✐♣♦t❡♥✉s❛ d✱ ❛r❡st❛ ❞♦ ♣r✐s♠❛✳ ❙❡ ♦ t❡♠♣♦ q✉❡ ❛ ❣♦t❛M, _{❧❡✈❛ ♣❛r❛ ♣❡r❝♦rr❡r ❛ ❞✐stâ♥❝✐❛}_d_{❝♦♠ ✈❡❧♦❝✐❞❛❞❡}

r❡❧❛t✐✈❛ Vr é T✱ ❡♥tã♦ s✉❛ ❝♦♠♣♦♥❡♥t❡ Vrn ❝♦♠ ♦ ♠❡s♠♦ t❡♠♣♦ T ♣❡r❝♦rr❡rá ✉♠❛

❞✐stâ♥❝✐❛ h✳ ❉✐❛♥t❡ ❞✐ss♦✱ ❛ ❛❧t✉r❛ ❞♦ ♣r✐s♠❛ s❡rá ❞❛❞❛ ♣♦r✿

h=T.||Vrn|| ✭✸✳✼✮

♦♥❞❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ✭ ✷✳✼ ✮✱ ♦ ♠ó❞✉❧♦ ❞❡ Vrn é ❞❛❞♦ ♣❡❧❛ s♦♠❛ ❞♦s ♠ó❞✉❧♦s ❞❛s ❝♦♠✲

♣♦♥❡♥t❡s ♥♦r♠❛✐s ❛♦ ♣❧❛♥♦ ❞❡B ❞❛s ✈❡❧♦❝✐❞❛❞❡s M(Vp) ❡M,(Vg)✳

❈á❧❝✉❧♦ ❞♦s ♠ó❞✉❧♦s ❞❛s ❝♦♠♣♦♥❡♥t❡s ♥♦r♠❛✐s ❛♦ ♣❧❛♥♦ ❞❡B❞❛s ✈❡❧♦❝✐❞❛❞❡sM(Vp)

❡ M,₍_V g)✳

P❛r❛ ✐ss♦✱ s❡rã♦ ❢❡✐t❛s ❛❧❣✉♠❛s ❥✉st✐✜❝❛t✐✈❛s✱ ❞❡ ❛❧❣✉♠❛s ♠❡❞✐❞❛s ❞❛ ✜❣✉r❛ ✸✳✺✳ Pr✐♠❡✐r❛ ❥✉st✐✜❝❛t✐✈❛

◆♦ tr✐â♥❣✉❧♦ GT I✱ ♦ â♥❣✉❧♦ IGTb =α✱ ♣♦✐sVp t❡♠ ❞✐r❡çã♦ ❤♦r✐③♦♥t❛❧✳

❙❡❣✉♥❞❛ ❥✉st✐✜❝❛t✐✈❛

❆❣♦r❛ ❝♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ GHF✱ r❡t♦ ❡♠ F✱ ♦♥❞❡||GF||=||Vpn|| é ♦ ♠ó❞✉❧♦ ❞❛

❝♦♠♣♦♥❡♥t❡ ❞❡ Vp ♥❛ ❞✐r❡çã♦ ♥♦r♠❛❧ ❛ B✳ ◆❡ss❡ tr✐â♥❣✉❧♦ t❡♠♦s q✉❡✿

FGHˆ = 90◦−α ❡ FGHˆ +GHFˆ +HF Gˆ = 180◦ ⇒90◦−α+GHFˆ +90◦ = 180◦ ⇒GHFˆ =α ❚❡r❝❡✐r❛ ❥✉st✐✜❝❛t✐✈❛

❖s tr✐â♥❣✉❧♦s GT I ❡ LJI sã♦ s❡♠❡❧❤❛♥t❡s✱ ♣♦rt❛♥t♦ ♦ â♥❣✉❧♦ JLIb = TGIb = α❀

❛❧é♠ ❞♦ ♠❛✐s✱ ♦ â♥❣✉❧♦SLRb =α−θ❀ ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ LRS✳ ❉❡ ♣♦ss❡ ❞❡ss❛s ❥✉st✐✜❝❛t✐✈❛s t❡♠♦s q✉❡✿

senα= ||Vpn||

||Vp||

⇒ ||Vpn||=||Vp||.senα ❡ cos(α−θ) =

||Vgn||

||Vg||

⇒ ||Vgn||=||Vg||.cos(α−θ)

(36)

✸✺

||Vrn||=||Vp||.senα+||Vg||.cos(α−θ) ✭✸✳✾✮

❉❛s ❡q✉❛çõ❡s (3.7) ❡(3.9)r❡s✉❧t❛♠ ❡♠

h=T.[||Vp||.senα+||Vg||.cos(α−θ)] ✭✸✳✶✵✮

P❡❧❛s ❡q✉❛çõ❡s (3.6)❡ (3.10) ❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ✈♦❧✉♠❡ ♣r♦❝✉r❛❞♦ é

V =A(B).h=A(B).T[||Vp||.senα+||Vg||.cos(α−θ)]

❆ss✐♠ ❝♦♠♦ ❢♦✐ ❢❡✐t♦ ♣❛r❛ ❛s ❡q✉❛çõ❡s ✭ ✸✳✹ ✮ ❡ ✭ ✸✳✺ ✮✱ ❢❛③❡♥❞♦ θ = 0✱ ♥❛ ❡q✉❛çã♦

(37)

(38)

✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s

❖ tr❛❜❛❧❤♦ t❡✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ❝❤✉✈❛ ✐♥❝✐❞❡♥t❡ ♥♦ ♣❛r❛✲❜r✐s❛ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✱ ♥✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ✜①♦✱ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❝♦♥st❛♥t❡ ❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❞❛s ❣♦t❛s ❞❡ ❝❤✉✈❛ ❝♦♥st❛♥t❡✳

P❛r❛ ✐ss♦✱ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❢♦✐ ❢❡✐t❛ ❡♠ ❞✉❛s ♣❛rt❡s✳

❆ ♣r✐♠❡✐r❛ ♣❛rt❡✱ ❝♦♥❝❧✉✐✉✲s❡ q✉❡ ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞❛ ❝❤✉✈❛ q✉❡ ✐♥❝✐❞❡ ♥♦ ♣❛r❛✲ ❜r✐s❛ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧ é ✉♠ ♣r✐s♠❛ ♦❜❧íq✉♦✳

❆ s❡❣✉♥❞❛ ♣❛rt❡✱ ❢♦✐ ❝❛❧❝✉❧❛❞♦ ♦ ✈♦❧✉♠❡ ❞♦ ♣r✐s♠❛ ❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❡❧❡ é ❞❡♣❡♥❞❡♥t❡ ❞❡ ❛❧❣✉♥s ♣❛râ♠❡tr♦s t❛✐s ❝♦♠♦✿ ❖ t❡♠♣♦ ❝♦♥s✐❞❡r❛❞♦ ❞❡ ♣❡r❝✉rs♦✱ ♦s â♥❣✉❧♦s ❞❡ ✐♥❝❧✐♥❛çã♦ ❞❛ ❝❤✉✈❛ ❝♦♠ r❡❧❛çã♦ à ✈❡rt✐❝❛❧ ❡ ❞♦ ♣❛r❛✲❜r✐s❛ ❝♦♠ r❡❧❛çã♦ à ❤♦r✐③♦♥t❛❧✱ ❡ ❞❛s ✈❡❧♦❝✐❞❛❞❡s ❞❛ ❝❤✉✈❛ ❡ ❞♦ ❛✉t♦♠ó✈❡❧✳

P♦rt❛♥t♦✱ ♣❛r❛ tr❛❜❛❧❤♦s ❢✉t✉r♦s ♥❡st❡ ♠❡s♠♦ ♣r♦❜❧❡♠❛❀ ♣♦❞❡ s❡r ❢❡✐t♦ ✉♠ ❡st✉❞♦ s♦❜r❡ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ✐♥❝❧✐♥❛çõ❡s ❞♦ ♣r✐s♠❛✱ ♦✉ ✉♠ ❡st✉❞♦ s♦❜r❡ ❝♦♠♦ ♠❛①✐♠✐③❛r ♦✉ ♠✐♥✐♠✐③❛r ❛ ❛r❡st❛ ❡ ♦ ✈♦❧✉♠❡ ❞♦ ♣r✐s♠❛✳ ❈♦♠ ✐ss♦✱ ❡st❛♠♦s ♠❛①✐♠✐③❛♥❞♦ ♦✉ ♠✐♥✐♠✐③❛♥❞♦ ♦ ✈♦❧✉♠❡ ❞❡ ❝❤✉✈❛ ✐♥❝✐❞❡♥t❡ ♥♦ ♣❛r❛ ❜r✐s❛ ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧✱ ❡♠ ✉♠ ♣❡r❝✉rs♦✱ ❞✉r❛♥t❡ ❞❡t❡r♠✐♥❛❞♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✳

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❘❡❢❡rê♥❝✐❛s

❬✶❪ ❘✳❈✳❇❆❙❙❆◆❊❩■✳ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❝♦♠ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ✸✳ ❡❞✳ ❙ã♦ P❛✉❧♦✿ ❈♦♥t❡①t♦✱ ✷✵✶✶✳

❬✷❪ ◆✳❈✳❏ú◆■❖❘✳ ❆ ❝❤✉✈❛ ❡ ♦ ♣❛r❛✲❜r✐s❛✳ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ✈✳ ✶✽✱ ♣✳ ✸✷✕✸✻✱ ✶✾✾✺✳

❬✸❪ ❍✳▼✳◆❯❙❙❊◆❩❱❊■●✳ ❈✉rs♦ ❞❡ ❋ís✐❝❛ ❇ás✐❝❛ ✲ ❱♦❧✉♠❡ ✶✳ ✹✳ ❡❞✳ ❙ã♦ P❛✉❧♦✿ ❊❞❣❛r❞ ❇❧✉❝❤❡r✱ ✷✵✵✷✳

❬✹❪ ❊✳▲✳▲■▼❆ ❡t ❛❧✳ ❆ ♠❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❱♦❧✉♠❡ ✶✳ ✻✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✵✻✳

❬✺❪ ❊✳▲✳▲■▼❆ ❡t ❛❧✳ ❆ ♠❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❱♦❧✉♠❡ ✷✳ ✻✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✵✻✳

❬✻❪ ❊✳▲✳▲■▼❆✳ ▼❡✉ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ ❡ ♦✉tr❛s ❤✐stór✐❛s✳ ✻✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✶✷✳

❬✼❪ ❈❆❘▼❖✱ ▼✳❀ ❊✳❲❆●◆❊❘❀ ❆✳❈✳▼❖❘●❆❉❖✳ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ◆ú♠❡r♦s ❝♦♠♣❧❡✲ ①♦s✳ ✸✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✵✻✳

❬✽❪ P✳❈✳P✳❈❆❘❱❆▲❍❖✳ ■♥tr♦❞✉çã♦ à ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✳ ✹✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱ ✷✵✵✺✳

❬✾❪ ❈❆▼❆❘●❖✱ ■✳❀ P✳❇❖❯▲❖❙✳ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳ ✸✳ ❡❞✳ ❙ã♦ P❛✉❧♦✿ P❡❛rs♦♥✱ ✷✵✵✺✳

(41)

(42)