Question

The speed $v$ of a particle moving along a straight line is given by $a+b v^2=x^2$ (where $x$ is its distance from the origin). The acceleration of the particle is

(A) $b x$

(B) $x / a$

(C) $x / b$

(D) $x / a b$

Answer 1

On differentiating w.r.t. $t$, we get

$\begin{array}{l}0+b\left(2 v \frac{d v}{d t}\right)=2 x \frac{d x}{d t} \\\Rightarrow \quad v b \frac{d v}{d t}=x \frac{d x}{d t} \\\Rightarrow \quad \frac{d v}{d t}=\frac{x}{v b} \cdot \frac{d x}{d t} \\\Rightarrow \quad \frac{d v}{d t}=\frac{x}{b} \quad\left(\because \frac{d x}{d t}=0\right) \\\end{array}$

So, The correct option will be (C).