Q.15 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 15 Solution

Topic: Motion in a Straight Line · Official NTA Booklet Code 45

Question

In some appropriate units, time (t) and position (x) relation of a moving particle is given by t = x² + x. The acceleration of the particle is

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Verified Against NTA Final Answer Key

Option (2) : -\frac{2}{(2x + 1)^3}

📚 Motion in a Straight Line Easy

📋 Given Parameters & Problem Setup

Time-Position Relation t = x² + x

Target Variable Acceleration of the particle a(x)

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The 30-Second Exam Hall Shortcut: ⚡ v(dv/dx) Chain Rule Shortcut

⏱️ Save ~40 seconds

dt/dx = 2x + 1 ⟹ v = (2x + 1)⁻¹ ⟹ a = v(dv/dx) = (2x + 1)⁻¹ · [-2(2x + 1)⁻²] = -2 / (2x + 1)³!

1 Differentiate given equation with respect to x: dt/dx = 2x + 1.

2 Velocity is the reciprocal of dt/dx: v = dx/dt = 1 / (2x + 1) = (2x + 1)⁻¹.

3 Acceleration formula in terms of position: a = v (dv/dx).

4 Differentiate v with respect to x: dv/dx = -2 (2x + 1)⁻².

5 Multiply: a = (2x + 1)⁻¹ × [-2 (2x + 1)⁻²] = -2 / (2x + 1)³ (or -2 v³).

6 Select Option (2) in under 15 seconds!

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Why You Can 100% Trust This Shortcut: Kinematics Calculus Relation (NCERT Physics · Motion in a Straight Line: a = v dv/dx).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Find Velocity as a Function of Position

Given the implicit trajectory relation t = x² + x, differentiate both sides with respect to time t using the chain rule:

\frac{dt}{dt} = 1 = \frac{d}{dt}(x^2 + x) = (2x + 1) \frac{dx}{dt} = (2x + 1) v \implies v = \frac{1}{2x + 1}

• v can be written in power notation: v = (2x + 1)⁻¹.

Step 2

Calculate Acceleration Using a = v (dv/dx)

Differentiate velocity with respect to position x:

\frac{dv}{dx} = \frac{d}{dx}(2x + 1)^{-1} = -1(2x + 1)^{-2} \cdot \frac{d}{dx}(2x + 1) = -\frac{2}{(2x + 1)^2}

• Now substitute into the kinematic identity for acceleration: a = v (dv/dx).

Step 3

Consolidate Acceleration Expression

Multiply velocity by its spatial derivative:

a = v \frac{dv}{dx} = \left(\frac{1}{2x + 1}\right) \left(-\frac{2}{(2x + 1)^2}\right) = -\frac{2}{(2x + 1)^3} = -2v^3

• The negative sign indicates retardation.
• Matches Option (2) perfectly.

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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 A

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Base has (x + 2) instead of (2x + 1).

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Student Fallacy (Why candidates pick this trap): Algebraic slip in the derivative of x² + x.

2 B

✓ 100% Correct

Verdict & Scientific Reason:

100% correct. a = v(dv/dx) = -2 / (2x + 1)³.

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Why this is airtight: None; textbook calculus.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Sign is positive and denominator is wrong.

⚠️

Student Fallacy (Why candidates pick this trap): Missing the negative sign from power rule and wrong base.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. This represents 2v, not acceleration.

⚠️

Student Fallacy (Why candidates pick this trap): Failing to take the second derivative.

⚠️

NTA Trap Alert: Missing Negative Sign & Power Slip

Differentiating (2x + 1)⁻¹ yields a negative sign: d/dx[(u)⁻¹] = -u⁻² u'. Forgetting the negative sign leads students to choose positive options.

🛡️ How to avoid the -1 mark: Power rule: exponent -1 brings down a factor of -1!

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Motion in a Straight Line Standard Benchmark

Core Scientific Principle Tested:

"Acceleration in terms of position is a = v (dv/dx) = -2v³."

Syllabus Focus: Kinematics of Non-Uniform Acceleration

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

t = x² + x ⟹ v = (2x + 1)⁻¹ ⟹ a = -2 / (2x + 1)³ = -2v³!

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NEET 2025 Physics Question 15 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ v(dv/dx) Chain Rule Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Find Velocity as a Function of Position

Calculate Acceleration Using a = v (dv/dx)

Consolidate Acceleration Expression

360° Option Elimination Engine Key Feature

NTA Trap Alert: Missing Negative Sign & Power Slip

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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