(B) We know that velocity $v = \frac{dx}{dt}$, so $\frac{dt}{dx} = \frac{1}{v}$.Now, differentiate with respect to $x$:$\frac{d^2 t}{dx^2} = \frac{d}{

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(B) We know that velocity $v = \frac{dx}{dt}$, so $\frac{dt}{dx} = \frac{1}{v}$.Now, differentiate with respect to $x$:$\frac{d^2 t}{dx^2} = \frac{d}{dx} \left( \frac{1}{v} \right) = -\frac{1}{v^2} \frac{dv}{dx}$.Using the chain rule, $\frac{dv}{dx} = \frac{dv}{dt} \times \frac{dt}{dx} = f \times \frac{1}{v} = \frac{f}{v}$.Substituting this back into the equation:$\frac{d^2 t}{dx^2} = -\frac{1}{v^2} \times \frac{f}{v} = -\frac{f}{v^3}$.Rearranging for $f$, we get $f = -v^3 \frac{d^2 t}{dx^2}$.

Class 12 Mathematics Application of Derivatives Rate of Change of Quantities

Medium

If the displacement, velocity, and acceleration of a particle at time $t$ are $x, v$, and $f$ respectively, then which of the following is true?

WBJEE 2010 All WBJEE

A
$f=v^3 \frac{d^2 t}{d x^2}$
B
$f=-v^3 \frac{d^2 t}{d x^2}$
C
$f=v^2 \frac{d^2 t}{d x^2}$
D
$f=-v^2 \frac{d^2 t}{d x^2}$

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Application of Derivatives

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Rate of Change of Quantities

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WBJEE 2010 Paper All WBJEE years →

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If the displacement, velocity, and acceleration of a particle at time $t$ are $x, v$, and $f$ respectively, then which of the following is true?

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$A$ closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After $21 \, min$,it was found that the height of the water column is half of the original height. How much more time in minutes does it require to empty the vessel?

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