(a) Let be the angle at which the projectile is fired w.r.t. the x-axis, since depends on t
Therefore tan since (Vy=Voy -gt and Vx = Vox)
(b) Since = sin2 /2g…….(1)
R = sin2 /g …….(2)
Dividing (1) by (2)
( /R) = [ sin2 /2g]/[ sin2 /g] = / 4
<p><strong>4.32</strong></p><p><strong>(a)</strong> Let <span title="Click to copy mathml"><math><mi mathvariant="bold-italic">θ</mi><mfenced separators="|"><mrow><mrow><mi mathvariant="bold-italic">t</mi></mrow></mrow></mfenced><mo>=</mo><mi mathvariant="bold-italic"></mi><mrow><mrow><msup><mrow><mrow><mi mathvariant="bold">tan</mi></mrow></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow></mrow></msup></mrow><mo>?</mo><mrow><mfenced separators="|"><mrow><mrow><mfrac><mrow><mrow><msub><mrow><mrow><mi mathvariant="bold-italic">v</mi></mrow></mrow><mrow><mrow><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">y</mi></mrow></mrow></msub><mo>-</mo><mi mathvariant="bold-italic"></mi><mi mathvariant="bold-italic">g</mi><mi mathvariant="bold-italic">t</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi mathvariant="bold-italic">v</mi></mrow></mrow><mrow><mrow><mi mathvariant="bold-italic">o</mi><mi mathvariant="bold-italic">x</mi></mrow></mrow></msub></mrow></mrow></mfrac></mrow></mrow></mfenced></mrow></mrow></math></span> be the angle at which the projectile is fired w.r.t. the x-axis, since <span title="Click to copy mathml"><math><msub><mrow><mrow><mi mathvariant="bold-italic">θ</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mo>=</mo><mi mathvariant="bold-italic"></mi><mrow><mrow><msup><mrow><mrow><mi mathvariant="bold">tan</mi></mrow></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow></mrow></msup></mrow><mo>?</mo><mrow><mfenced separators="|"><mrow><mrow><mfrac><mrow><mrow><mn>4</mn><msub><mrow><mrow><mi mathvariant="bold-italic">h</mi></mrow></mrow><mrow><mrow><mi mathvariant="bold-italic">m</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi mathvariant="bold-italic">R</mi></mrow></mrow></mfrac></mrow></mrow></mfenced></mrow></mrow></math></span> depends on t</p><p>Therefore tan <span title="Click to copy mathml"><math><mi>θ</mi></math></span> since (Vy=Voy -gt and Vx = Vox)</p><p><strong>(b)</strong> Since <span title="Click to copy mathml"><math><mi>θ</mi></math></span> = <span title="Click to copy mathml"><math><mi>θ</mi><mfenced separators="|"><mrow><mrow><mi>t</mi></mrow></mrow></mfenced><mo>=</mo><mfrac><mrow><mrow><mi>V</mi><mi>x</mi></mrow></mrow><mrow><mrow><mi>V</mi><mi>y</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mi>V</mi><mi>o</mi><mi>y</mi><mo>-</mo><mi>g</mi><mi>t</mi></mrow></mrow><mrow><mrow><mi>V</mi><mi>o</mi><mi>x</mi></mrow></mrow></mfrac><mi></mi></math></span> sin2 <span title="Click to copy mathml"><math><mi>θ</mi><mfenced separators="|"><mrow><mrow><mi>t</mi></mrow></mrow></mfenced><mo>=</mo><mi></mi><mrow><mrow><msup><mrow><mrow><mi mathvariant="normal">tan</mi></mrow></mrow><mrow><mrow><mo>-</mo><mn>1</mn></mrow></mrow></msup></mrow><mo>?</mo><mrow><mo>(</mo><mfrac><mrow><mrow><mi>V</mi><mi>o</mi><mi>y</mi><mo>-</mo><mi>g</mi><mi>t</mi></mrow></mrow><mrow><mrow><mi>V</mi><mi>o</mi><mi>x</mi></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> /2g…….(1)</p><p>R = <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></mrow></msub></math></span> sin2 <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>u</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi></mi></mrow></mrow></msup></math></span> /g …….(2)</p><p>Dividing (1) by (2)</p><p>( <span title="Click to copy mathml"><math><mi>θ</mi></math></span> /R) = [ <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>u</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span> sin2 <span title="Click to copy mathml"><math><mi>θ</mi></math></span> /2g]/[ <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></mrow></msub></math></span> sin2 <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>u</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi></mi></mrow></mrow></msup></math></span> /g] = <span title="Click to copy mathml"><math><mi>θ</mi></math></span> / 4</p>
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