4.14 In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?

4.14

The velocity of the boat $V_b$ = 51 km/h, the velocity of the wind $V_w$ = 72 km/h

Flag is fluttering in N-E direction, so the $V_w$ direction is N-E.

When the ship begins to sail towards North, the flag will flutter along the direction of relative velocity of the wind w.r.t. the boat.

The angle between $V_w$ & - $V_b$ = 90 $°+45°$

tan $\beta$ = $\frac{V_{b}sin(90+45)}{V_w+V_{b}cos(90+45)}$ = $\frac{51\mathrm{s}\mathrm{i}\mathrm{n}?(90+45)}{72+51\mathrm{c}\mathrm{o}\mathrm{s}?(90+45)}$ = $\frac{36.062}{35.937}$

$\beta =$ tan-1( $\frac{36.062}{35.937}$ ) = 45.099 $°$

Angle with respect to East direction = 45.099 $°$ - 45 $°$ = 0.099 $°$

Hence the flag will flutter almost due East.

<p><strong>4.14</strong></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728973416phpbCvMeq_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728973416phpbCvMeq.jpeg" alt="" width="314" height="282"></picture></div></div><p>The velocity of the boat&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub></math></span>&nbsp;= 51 km/h, the velocity of the wind&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub></math></span>&nbsp;= 72 km/h</p><p>Flag is fluttering in N-E direction, so the&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub></math></span>&nbsp;direction is N-E.</p><p>When the ship begins to sail towards North, the flag will flutter along the direction of relative velocity of the wind w.r.t. the boat.</p><p>The angle between&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub></math></span>&nbsp;&amp; -&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub></math></span>&nbsp;= 90&nbsp;<span title="Click to copy mathml"><math><mo>°</mo><mo>+</mo><mn>45</mn><mo>°</mo></math></span></p><p>tan&nbsp;<span title="Click to copy mathml"><math><mi></mi><mi>β</mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub><mi>s</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>90</mn><mo>+</mo><mn>45</mn><mo>)</mo></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub><mo>+</mo><msub><mrow><mrow><mi>V</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub><mi>c</mi><mi>o</mi><mi>s</mi><mo>(</mo><mn>90</mn><mo>+</mo><mn>45</mn><mo>)</mo></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>51</mn><mi></mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo>?</mo><mo>(</mo><mn>90</mn><mo>+</mo><mn>45</mn><mo>)</mo></mrow></mrow><mrow><mrow><mn>72</mn><mo>+</mo><mn>51</mn><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo>?</mo><mo>(</mo><mn>90</mn><mo>+</mo><mn>45</mn><mo>)</mo></mrow></mrow></mfrac><mi></mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>36.062</mn></mrow></mrow><mrow><mrow><mn>35.937</mn></mrow></mrow></mfrac></math></span></p><p><span title="Click to copy mathml"><math><mi>β</mi><mo>=</mo><mi></mi></math></span>&nbsp;tan-1(&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>36.062</mn></mrow></mrow><mrow><mrow><mn>35.937</mn></mrow></mrow></mfrac></math></span>&nbsp;) = 45.099&nbsp;<span title="Click to copy mathml"><math><mo>°</mo></math></span></p><p>Angle with respect to East direction = 45.099&nbsp;<span title="Click to copy mathml"><math><mo>°</mo></math></span>&nbsp;- 45&nbsp;<span title="Click to copy mathml"><math><mo>°</mo></math></span>&nbsp;= 0.099&nbsp;<span title="Click to copy mathml"><math><mo>°</mo></math></span></p><p>Hence the flag will flutter almost due East.</p>

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