WBJEE · Maths · Application of Derivatives
If the displacement, velocity and acceleration of a particle at time, \(\mathrm{t}\) be \(\mathrm{x}, \mathrm{v}\) and \(\mathrm{f}\) respectively, then which one is true?
- A \(\mathrm{f}=\mathrm{v}^3 \frac{\mathrm{d}^2 t}{d \mathrm{x}^2}\)
- B \(f=-v^3 \frac{d^2 t}{d x^2}\)
- C \(\mathrm{f}=\mathrm{v}^2 \frac{\mathrm{d}^2 \mathrm{t}}{\mathrm{dx}^2}\)
- D \(f=-v^2 \frac{d^2 t}{d x^2}\)
Answer & Solution
Correct Answer
(B) \(f=-v^3 \frac{d^2 t}{d x^2}\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & \text { Hints: } \frac{d^2 t}{d x^2}=\frac{d\left(\frac{d t}{d x}\right)}{d x}=\frac{d\left(\frac{1}{v}\right)}{d x}=-\frac{1}{v^2} \frac{d v}{d t} \times \frac{1}{v} \\ & \Rightarrow f=-v^3 f \frac{d^2 t}{d x^2} \end{aligned} \]
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