First slide

Instantaneous acceleration

Question

A point moves in a straight line so that its displacement x m at time t sec is given by $x^{2} = 1 + t^{2} .$ Its acceleration in m/sec2 at a time t sec is

Moderate

A

$1 / x^{3}$
B

$t / x^{3}$
C

$\frac{1}{x} - \frac{t^{2}}{x^{3}}$
D

$\frac{1}{x} - \frac{1}{x^{2}}$

Solution

$\begin{array}{l} x^{2} = 1 + t^{2} (x^{2} - t^{2}) = 1 \\ 2 x (\frac{dx}{dt}) = 2 t . \\ x_{u} = t \\ \frac{dx}{dt} v + x (\frac{d v}{dt}) = 1 \\ v^{2} + ax = 1 . \\ ax = 1 - v^{2} \\ a = \frac{1 - v^{2}}{x} = \frac{1}{x} - \frac{t^{2}}{x^{2} (x)} = \frac{x^{2} - t^{2}}{x^{3}} = \frac{1}{x^{3}} \end{array}$

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