Q. The coordinates of a moving particle at any time ‘t’ are given by x = αt^{3} and y = βt^{3}. The speed of the particle at time ‘t’ is given by

(a) $ \displaystyle \sqrt{\alpha^2 + \beta^2} $

(b) $ \displaystyle 3t \sqrt{\alpha^2 + \beta^2} $

(c) $ \displaystyle 3t^2 \sqrt{\alpha^2 + \beta^2} $

(d) $\displaystyle t^2 \sqrt{\alpha^2 + \beta^2} $

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Differentiating w.r.t time

$ \displaystyle \frac{dx}{dt} = 3\alpha t^2 $

$ \displaystyle v_x = 3\alpha t^2 $

y = βt^{3}

$\displaystyle \frac{dy}{dt} = 3\beta t^2 $

$ \displaystyle v_y = 3\beta t^2 $

Speed of particle is

$ \displaystyle v = \sqrt{v_x ^2 + v_y ^2} $

$ \displaystyle v = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2} $

$ \displaystyle = 3t^2 \sqrt{\alpha^2 + \beta^2} $