Acceleration vs Distance – Find the Power of x (n) | JEE One-Dimensional Motion Doubt | Doubtify JEE

 

📈 Acceleration as a Function of Distance – Motion in One Dimension | JEE Physics | Doubtify JEE

🧮 Question:

The distance x covered by a particle in one-dimensional motion varies with time t as:
x² = at² + 2bt + c.
If the acceleration of the particle depends on x as x⁻ⁿ, where n is an integer, what is the value of n?

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🖼️ Question Image:

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✅ Detailed Concept & Solution:

This is a JEE-level conceptual problem involving the relation between distance, velocity, and acceleration — with a twist of functional dependence.

We are given:
x² = at² + 2bt + c — (1)
To find acceleration and its dependency on x, let’s differentiate both sides of equation (1):

Differentiate w.r.t. time t:

2x(dx/dt) = 2at + 2b
⇒ v = dx/dt = (at + b)/x — (2)

Now differentiate velocity v w.r.t time to find acceleration:

a = dv/dt = d/dt[(at + b)/x]

Use quotient rule:

a = [a·x - (at + b)(dx/dt)] / x²
Plug dx/dt = (at + b)/x from (2):
a = [a·x - (at + b)·((at + b)/x)] / x²
⇒ a = [a·x² - (at + b)²] / x³ — (3)

Now observe the form: acceleration a ∝ 1/x³,
So the acceleration is proportional to x⁻³.

Hence, n = 3

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🎯 Final Answer:

The value of n is 3.

🖼️ Solution Image:

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🎥 Video Solution:

🔍 Why this Question is Important:

• Combines kinematics with functional calculus – a unique skill tested in JEE.

• Requires understanding of implicit differentiation, chain rule, and functional dependency.

• Helps you master acceleration–displacement type problems, which appear often in advanced exams.

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💡 Pro Tip:

Always be ready to differentiate composite functions when distance or position is not linearly related to time. Questions like this often look complex, but simplifying one layer at a time leads to the answer.

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