TS EAMCET · Physics · Motion In Two Dimensions

The path of a projectile is given by the equation \(y=a x-b x^2\), where \(a\) and \(b\) are constants and \(x\) and \(y\) are respectively horizontal and vertical distances of projectile from the point of projection. The maximum height attained by the projectile and the angle of projection are respectively

A \(\frac{2 a^2}{b}, \tan ^{-1}(a)\)
B \(\frac{b^2}{2 a}, \tan ^{-1}(b)\)
C \(\frac{a^2}{b}, \tan ^{-1}(2 b)\)
D \(\frac{a^2}{4 b}, \tan ^{-1}(a)\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{a^2}{4 b}, \tan ^{-1}(a)\)

Step-by-step Solution

Detailed explanation

The given equation, \[ y=a x-b x^2 \] and we know that equation of trajectory is \[ y=(\tan \theta) x-\frac{1}{2} \frac{g}{u^2 \cos ^2 \theta} \cdot x^2 \] Compare both equations, we get \[ a=\tan \theta, b=\frac{1}{2} \cdot \frac{g}{u^2 \cos ^2 \theta} \] The maximum height…

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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