JEE Mains · Physics · STD 11 - 3.2 motion in plane

A projectile is projected with velocity of \(25\, m / s\) at an angle \(\theta\) with the horizontal. After t seconds its inclination with horizontal becomes zero. If \(R\) represents horizontal range of the projectile, the value of \(\theta\) will be : [use \(g =10 m / s ^{2}\) ]

A \(\frac{1}{2} \sin ^{-1}\left(\frac{5 t^{2}}{4 R}\right)\)
B \(\frac{1}{2} \sin ^{-1}\left(\frac{4 R }{5 t ^{2}}\right)\)
C \(\tan ^{-1}\left(\frac{4 t ^{2}}{5 R }\right)\)
D \(\cot ^{-1}\left(\frac{ R }{20 t ^{2}}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\cot ^{-1}\left(\frac{ R }{20 t ^{2}}\right)\)

Step-by-step Solution

Detailed explanation

\(R =\frac{ V ^{2}(2 \sin \theta \cos \theta)}{ g }\) \(t =\frac{ V \sin \theta}{ g } \Rightarrow V =\frac{ gt }{\sin \theta}\) \(\Rightarrow R =\frac{ g ^{2} t ^{2}}{\sin ^{2} \theta} \cdot \frac{2 \sin \theta \cos \theta}{ g }\)…

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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Match List \(I\) with List \(II\)

List \(I\)	List \(II\)
\(A\) Spring constant	\(I\) \((T ^{-1})\)
\(B\) Angular speed	\(II\) \((MT ^{-2})\)
\(C\) Angular momentum	\(III\) \((ML ^2)\)
\(D\) Moment of Inertia	\(IV\) \((ML ^2 T ^{-1})\)

Choose the correct answer from the options given below

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