Q.8 Physics ⭐ 360° Option Elimination Active

NEET 2021 Physics Question 8 Solution

Topic: Motion in a Straight Line · Official NTA Booklet Code M1

Question

A small block slides down on a smooth inclined plane, starting from rest at time t = 0. Let Sₙ be the distance travelled by the block in the interval t = n − 1 to t = n. Then, the ratio Sₙ/Sₙ₊₁ is:

✓

Verified Against NTA Final Answer Key

Option (2) : (2n - 1) / (2n + 1)

📚 Motion in a Straight Line Easy

📋 Given Parameters & Problem Setup

Initial Velocity (u) 0 (starts from rest)

Acceleration (a) g sin θ (constant down smooth incline)

Distance S_n Distance in interval t = n - 1 to t = n (n-th second)

Distance S_{n+1} Distance in interval t = n to t = n + 1 ((n+1)-th second)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Galileo's Odd Number Rule for nth Second

⏱️ Save ~20 seconds

For any motion from rest under constant acceleration, distance in successive seconds follows odd integers: $S_n \propto (2n - 1)$ . For the next second, replace $n$ with $n+1$ : $S_{n+1} \propto 2(n+1) - 1 = 2n + 1$ . The ratio is directly $\frac{2n - 1}{2n + 1}$ ! Match Option (2)!

1 Standard formula for distance in the

n

-th second starting from rest (

u = 0

S_n = \frac{a}{2}(2n - 1)

2 For

(n+1)

-th second, replace

n

with

(n+1)

S_{n+1} = \frac{a}{2}[2(n+1) - 1] = \frac{a}{2}(2n + 1)

3 Take the ratio:

\frac{S_n}{S_{n+1}} = \frac{\frac{a}{2}(2n - 1)}{\frac{a}{2}(2n + 1)} = \frac{2n - 1}{2n + 1}

4 Match Option (2).

🛡️

Why You Can 100% Trust This Shortcut: Canonical kinematic equation for distance in the nth unit time interval.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Formula for Distance Travelled in the n-th Second

For an object moving with initial velocity $u$ and uniform acceleration $a$ , the distance covered between time $t = n-1$ and $t = n$ is:

S_n = S(n) - S(n-1) = u + \frac{a}{2}(2n - 1)

Step 2

Applying Initial Conditions

Since the block starts from rest on a smooth inclined plane of angle $\theta$ , $u = 0$ and $a = g\sin\theta$ (constant):

S_n = \frac{a}{2}(2n - 1)

Step 3

Finding S_{n+1} and Computing the Ratio

In the interval $t = n$ to $t = n+1$ , the distance travelled is $S_{n+1}$ :

S_{n+1} = \frac{a}{2}\left[2(n+1) - 1\right] = \frac{a}{2}(2n + 1)

Step 4

Final Ratio

Dividing $S_n$ by $S_{n+1}$ cancels the common acceleration factor $a/2$ :

\frac{S_n}{S_{n+1}} = \frac{2n - 1}{2n + 1}

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 1

✗ Incorrect

Verdict & Scientific Reason:

Incorrect denominator: evaluates $S_{n+1}$ erroneously as $2n$ .

⚠️

Student Fallacy (Why candidates pick this trap): Writing

2n

instead of

2(n+1)-1 = 2n+1

2 2

✓ 100% Correct

Verdict & Scientific Reason:

Exact ratio of distances in $n$ -th and $(n+1)$ -th intervals.

💡

Why this is airtight: None; textbook kinematic derivation.

3 3

✗ Incorrect

Verdict & Scientific Reason:

Inverted ratio ( $S_{n+1} / S_n$ ).

⚠️

Student Fallacy (Why candidates pick this trap): Inverting numerator and denominator.

4 4

✗ Incorrect

Verdict & Scientific Reason:

Inverted and miscalculated ratio.

⚠️

Student Fallacy (Why candidates pick this trap): Confusing total distance up to time

n

with distance in

n

-th second.

⚠️

NTA Trap Alert: Total Distance vs. Distance in nth Second Trap

Students frequently confuse total distance in $n$ seconds ( $S = \frac{1}{2}at^2 \propto n^2$ ) with distance in the $n$ -th second ( $S_n = \frac{a}{2}(2n-1)$ ).

🛡️ How to avoid the -1 mark: The interval from

n-1

n

is the

n

-th second:

S_n = \frac{a}{2}(2n-1)

📖

Core NEET Syllabus Standard

NCERT Physics · Motion in a Straight Line Standard Benchmark

Core Scientific Principle Tested:

"Distance travelled in the $n$ -th second of uniformly accelerated motion from rest is $S_n = \frac{a}{2}(2n - 1)$ ."

Syllabus Focus: Kinematic Equations for Uniformly Accelerated Motion

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2021 Physics Question 8 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Galileo's Odd Number Rule for nth Second

Step-by-Step Rigorous Workflow & Mathematical Proof

Formula for Distance Travelled in the n-th Second

Applying Initial Conditions

Finding S_{n+1} and Computing the Ratio

Final Ratio

360° Option Elimination Engine Key Feature

NTA Trap Alert: Total Distance vs. Distance in nth Second Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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