Question

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

• 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. $f=-v^2\frac{d^{2}t}{d{x}^2}$

Answer 1

The correct option is B $f=-v^3\frac{d^{2}t}{d{x}^2}$

We know that the velocity can be written as:

$v=\frac{dx}{dt}$

Therefore, $\frac{1}{v}=\frac{dt}{dx}$

Now, expressing $f$ in terms of velocity.

$\frac{d^{2}t}{d{x}^2}=\frac{d\left(\begin{array}{c}\frac{dt}{dx}\end{array}\right)}{dx}=\frac{d\left(\begin{array}{c}\frac{1}{v}\end{array}\right)}{dx}=-\frac{1}{v^2}\frac{dv}{dt}\times \frac{1}{v}$
$\Rightarrow f=-v^3\frac{d^{2}t}{d{x}^2}$