Kinematics NEET 2021: Find θ for Max Height 4R with Time T

What 2021 NEET kinematics question links circular revolution time T to a projectile height of exactly 4R?

A particle completes one revolution in radius R in time T. It uses the same speed for projection at angle θ to the horizontal, resulting in maximum height exactly 4R. The task is to identify the correct inverse trigonometric expression for θ among four options. The correct expression is option A.

θ = \sin^{- 1} [{(\frac{2 g T^{2}}{π^{2} R})}^{1 / 2}]

By following the official derivation and substitution checklist below, you can expect to solve this exact 2021 question in one pass because it isolates the inversion error that produces distractors.

How does the official method solve this kinematics NEET 2021 projectile and circular motion question?

The official method starts from u derived from uniform circular motion, sets maximum height to 4R, and solves for θ without early numerical substitution.

u = \frac{2 π R}{T}

u^{2} = \frac{4 π^{2} R^{2}}{T^{2}}

Maximum height is

H = \frac{u^{2} \sin^{2} θ}{2 g} .

Set equal to 4R:

\frac{u^{2} \sin^{2} θ}{2 g} = 4 R .

Rearrange to

\sin^{2} θ = \frac{8 g R}{u^{2}} .

Substitute u²:

\sin^{2} θ = 8 g R \times \frac{T^{2}}{4 π^{2} R^{2}} = \frac{2 g T^{2}}{π^{2} R} .

Take square root and inverse sine:

θ = \sin^{- 1} {(\frac{2 g T^{2}}{π^{2} R})}^{1 / 2} .

This matches option A exactly. Every substitution follows the verified official derivation.

What algebraic rearrangement error in this kinematics NEET 2021 question produces one of the wrong options?

The precise mistake is inverting the height equation at isolation and writing sin²θ = u²/(8gR) instead of 8gR/u². After inserting u² = 4π²R²/T² this yields argument π²R/(2gT²). Taking the square root and then inverse cosine, which confuses the vertical component, produces a form matching option C or D.

This inversion in formula rearrangement is the single concrete error that generates the distractor. Spot it first and the remaining choices become straightforward.

Which core kinematics concepts connect uniform circular motion speed to projectile maximum height?

Linear speed in uniform circular motion is exactly 2πR/T, not πR/T. Maximum height depends solely on the vertical component squared: (u sinθ)²/2g.

The height formula uses sin²θ while range uses sin2θ. The argument of sin⁻¹ must be dimensionless. The combination gT²/R has dimensions of length × time⁰ divided by length and cancels cleanly, confirming the expression inside the square root is physically valid.

What similar practice problems combine circular motion parameters with projectile quantities?

These three problems repeat the exact linkage of circular speed u = 2πR/T to projectile height or time of flight.

A particle moving in a circle of radius R with period T/2 is projected with the same speed at angle θ to the horizontal. If the maximum height attained equals 2R, find θ.
The maximum height equals the radius R when the projection speed is derived from circular motion of time period T. Obtain the expression for cosθ.
Using the same u from circular motion of period T that gave maximum height 4R, find the time of flight for the projectile and compare its magnitude with T.

Apply the official substitution checklist to each. You can expect accuracy on future hybrid questions to improve because the algebraic pattern is identical.

To locate every past kinematics NEET 2021 question plus its worked solution, search the past-paper archive by year and chapter. It returns the exact official method every time.

How should you tackle hybrid kinematics PYQs on exam day inside the 180-minute NEET paper?

Write u = 2πR/T on first line before touching projectile formula. Never insert numerical g or π until final step; keep symbolic. Verify final argument of inverse trig is less than 1 for physical sense. Expected solving time 120 s; allocate no more than 150 s.

Mark the answer and move on. The remaining time is better spent on the other physics questions that reward the same symbolic habit.

Next step: photograph a doubt on NEET JEEnius AI and photograph any question you are stuck on and get a step-by-step solution across Physics, Chemistry and Biology (20 free a month).

Keep going with Gene Cloning Tools NEET 2025: Why C and D Are Correct.

Frequently asked questions

What is the correct expression for theta in the kinematics NEET 2021 question?

The correct expression is θ = sin^{-1} [(2gT²/π²R)^{1/2}]. Start from u = 2πR/T obtained from circular motion, set maximum height u²sin²θ/2g equal to 4R and solve for sinθ after substitution. This matches option A in the 2021 paper.

How do you derive the angle theta in the NEET 2021 kinematics circular projectile problem?

Write u = 2πR/T and square it to get u² = 4π²R²/T². Set H = u² sin²θ / 2g = 4R and rearrange to sin²θ = 8gR/u². Substitute u² to obtain sin²θ = 2gT²/π²R. Take square root and apply sin^{-1} to reach the final expression.

What is the common mistake in the kinematics NEET 2021 question on projectile height?

The frequent error is inverting the height equation and writing sin²θ = u²/(8gR) instead of the correct 8gR/u². After substituting u² = 4π²R²/T² this produces the reciprocal argument π²R/(2gT²). Students then sometimes switch to inverse cosine, matching the distractor options.

What is the speed u in the kinematics NEET 2021 circular motion to projectile question?

The speed is exactly u = 2πR/T. One revolution covers circumference 2πR in time T, so linear speed equals distance over time. This u is then used directly in the projectile maximum height formula without any numerical substitution until the end.

How to solve hybrid kinematics NEET questions on exam day?

Always start by writing u = 2πR/T on the first line. Keep the entire derivation symbolic and substitute only at the end. Verify the argument of sin^{-1} is dimensionless and less than 1. Allocate at most 150 seconds and move on after marking the answer.

Kinematics NEET 2021: θ from Circular T to Height 4R