Question

a body is projected with velocity u such that its horizontal range and maximum vertical heights are same the maximum heights is
(1)u^2/2g
(2)3u^2/4g
(3)16u^2/17g
(4)8u^2/17g​

Answer 1

Answer :-

Given :-

Inital velocity = u

Horizontal range = Maximum height

To find :-

Maximum height

Formula used :-

➱ $\rm Maximum \:height = \frac{u^2 sin^2 \theta}{2g}$

➱ $\rm Horizontal \:range = \frac{u^2 sin2 \theta}{g}$

➱ $\rm sec^2 \theta = 1 + tan^2 \theta$

➱ $\rm sin^2\theta + cos^2\theta = 1$

Solution :-

➣ Equating the formulas of maximum height and horizontal range :-

➱ $\rm \frac{\cancel {u^2} sin^2 \theta}{2 \cancel g} = \frac{\cancel {u^2} sin2 \theta}{\ cancel g}$

➱ $\rm sin^2 \theta = sin 2 \theta$

➱ $\rm \cancel {sin^ 2 \theta} = 4 \cancel {sin \theta} cos\theta$

↬ $\rm\red{ sin\theta = 4 cos\theta }$

➱ $\rm \frac{sin\theta }{cos\theta} = 4$

↬ $\rm tan \theta = 4$

➣ By using the trigonometric identity :-

➯ $\rm sec^2 \theta = 1 + tan^2 \theta$

➯ $\rm \frac{1}{cos^2\theta} = 1 + 4^2$

➯ $\rm \frac{1}{cos^2\theta} = 17$

↬ $\rm\red{ cos^2\theta = \frac{1}{17}}$

➯ $\rm sin^2\theta + cos^2\theta = 1$

➯ $\rm sin^2\theta = 1 - \frac{1}{17}$

↬ $\rm\red{ sin^2\theta = \frac{16}{17}}$

➣ Calculating Maximum height

• Initial velocity = u
• $\rm sin^2\theta = \frac{16}{17}$

➯ $\rm Maximum\: height = u^2sin^2\theta$

➯ $\rm H = \Big(\frac{u^2}{2g}\Big) \frac{16}{17}$

↬ $\boxed{\rm\red{ H = \frac{8u^2}{17g}}}$

➣ Additional information :-

$\rm Time\: of \:flight = \frac{2usin\theta}{g}$

$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$

Answer 2

Answer:

option 4

8u^2/17g

Explanation:

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