MHT CET · Physics · Oscillations

A particle performs linear SHM. At a particular instant, the velocity of the particle is $u$ and acceleration is $\alpha$ (both having the same direction). At another instant velocity is $v$ and acceleration is $\beta(0<\alpha<\beta)$ (both in opposite direction to each other). The distance between the two positions is

A $\frac{u^{2} - v^{2}}{α + β}$
B $\frac{u^{2} + v^{2}}{α + β}$
C $\frac{u^{2} - v^{2}}{α - β}$
D $\frac{u^{2} + v^{2}}{α - β}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{u^{2} - v^{2}}{α + β}$

Step-by-step Solution

Detailed explanation

Let distance be

x_{1}

when velocity is u and acceleration

α .

Let distance be

x_{2}

when velocity is v and acceleration

β .

ω

is the angular frequency then

α = ω^{2} x_{1}

And

β = ω^{2} x_{2}

∴ α + β = ω^{2} (x_{1} + x_{2})

....... (i)
Also

u^{2} = ω^{2} A^{2} - ω^{2} x_{1}^{2}

And

v^{2} = ω^{2} A^{2} - ω^{2} x_{2}^{2}

v^{2} - u^{2} = ω^{2} (x_{1}^{2} - x_{2}^{2})

v^{2} - u^{2} = ω^{2} (x_{1} - x_{2}) (x_{1} + x_{2})

........... (ii)
By Equation. (i) we get

v^{2} - u^{2} = (x_{1} - x_{2}) (α + β)

∴ x_{1} - x_{2} = \frac{v^{2} - u^{2}}{α + β}

x_{2} - x_{1} = \frac{u^{2} - v^{2}}{α + β}

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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