Question

The relation between time and distance is $t=\alpha x^2+\beta x$ where $\alpha$ and $\beta$ are constants. The retardation is

• A. $2\alpha v^3$
• B. $2\beta v^3$
• C. $2\alpha \beta v^3$
• D. $2{\beta }^2{v}^3$

Answer 1

Answer: (A)

$2\alpha v^3$

$\overrightarrow{v}=\frac{d\overrightarrow{x}}{dt}$
and
$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$

$t=\alpha x^2+\beta x$

Differentiating with respect to time on both sides, we get
$1=2\alpha \frac{dx}{dt}x+\beta \frac{dx}{dt}$

$\therefore v=\frac{1}{\beta +2\alpha x}$

$\frac{dv}{dt}=\frac{-2\alpha v}{(\beta +2\alpha x{)}^2}=-2\alpha v^3$

Negative sign shows retardation.
Option A is correct answer.