First slide

Rectilinear Motion

Question

The displacement x of a particle moving in a straight line varies with time t as $x^{2} = a + {bt}^{2}$ where a and b are constants. The acceleration of the particle at time t is given by

Difficult

A

$\frac{a}{x} - \frac{{bt}^{2}}{x^{3}}$
B

$\frac{a}{x} - \frac{bt}{x^{2}}$
C

$\frac{b}{x} - \frac{b^{2} t^{2}}{x^{3}}$
D

$\frac{a}{x} - \frac{at}{x^{2}}$

Solution

$x^{2} = a + {bt}^{2}$ .
Differentiating w.r.t time t,
$2 x \frac{dx}{dt} = 2 bt \Rightarrow \frac{dx}{dt} = \frac{bt}{x} \Rightarrow v = \frac{bt}{x}$
Acceleration
$= \frac{dv}{dt} = \frac{d}{dt} (\frac{bt}{x}) = \frac{b}{x} - \frac{bt}{x^{2}} \frac{dx}{dt} = \frac{b}{x} - \frac{bt}{x^{2}} \frac{bt}{x} = \frac{b}{x} (1 - \frac{{bt}^{2}}{x^{2}})$

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