Q. The coordinates of a moving particle at any time ‘t’ are given by x = αt3 and y = βt3. 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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Ans: (c)
Sol: x = αt3
Differentiating w.r.t time
$ \displaystyle \frac{dx}{dt} = 3\alpha t^2 $
$ \displaystyle v_x = 3\alpha t^2 $
y = βt3
$\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} $