Q57 / 155 · WBJEE 2018 · Mathematics

The law of motion of a body moving along a straight line is $x=\frac{1}{2}$ vt. $x$ being its distance from a fixed point on the line at time $t$ and $v$ is its velocity there, Then

  • A
    acceleration $f$ varies directly with $x$
  • B
    acceleration $f$ vares inversely with $x$
  • Correct answer
    C
    acceleration $f$ is constant
  • D
    acceleration $f$ varies directly with t

Solution

We have, $\quad x=\frac{1}{2} v t$ $\Rightarrow \quad x=\frac{1}{2} \frac{d x}{d t} t$ $\left[\because v=\frac{d x}{d t}\right]$ $\Rightarrow \quad \frac{2 d t}{t}=\frac{d x}{x}$ $\Rightarrow \quad 2 \cdot \int \frac{d t}{t}=\int \frac{d x}{x}$ $\Rightarrow \quad 2 \log |t|+\log |c|=\log |x|$ $\Rightarrow \quad \log \left(t^{2} \cdot c\right)=\log x$ $\Rightarrow \quad x=t^{2} c$ $\Rightarrow$ $\frac{d x}{d t}=2 t c$ $\Rightarrow$ $\frac{d^{2} x}{d t^{2}}=2$ $\Rightarrow$ acceleration $f$ is constant.

Hence, the correct answer is option: 'C'.

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