ઉર્ધ્વ સમતલમાં પ્રક્ષિપ્ત નો ગતિપથ \(y =\alpha x -\beta x ^{2}\) છે, જ્યાં \(\alpha\) અને \(\beta\) અચળાંકો છે તેમજ \(x\) અને \(y\) પ્રક્ષિપ્ત બિંદુ થી અનુક્રમે સમક્ષિતીજ અને ઉર્ધ્વ અંતર દર્શાવે છે. અહિંયા પ્રક્ષિપ્તકોણ \(\theta\) અને પ્રાપ્ત કરેલ મહત્તમ ઊંચાઈ \(H\) અનુક્રમે નીચે જણાવેલ વિકલ્પો દ્વારા દર્શાવામાં આવે છે :

A \(\tan ^{-1} \alpha, \frac{\alpha^{2}}{4 \beta}\)
B \(\tan ^{-1} \beta, \frac{\alpha^{2}}{2 \beta}\)
C \(\tan ^{-1} \alpha, \frac{4 \alpha^{2}}{\beta}\)
D \(\tan ^{-1}\left(\frac{\beta}{\alpha}\right), \frac{\alpha^{2}}{\beta}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\tan ^{-1} \alpha, \frac{\alpha^{2}}{4 \beta}\)

Step-by-step Solution

Detailed explanation

\(y =\alpha x -\beta x ^{2}\) comparing with trajectory equation \(y = x \tan \theta-\frac{1}{2} \frac{ gx ^{2}}{ u ^{2} \cos ^{2} \theta}\) \(\tan \theta=\alpha \Rightarrow \theta=\tan ^{-1} \alpha\) \(\beta=\frac{1}{2} \frac{ g }{ u ^{2} \cos ^{2} \theta}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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