The coordinates of a moving particle at any t | vedclass

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(C) Given the position coordinates of the particle as $x = \alpha t^3$ and $y = \beta t^3$.The velocity components are obtained by differentiating the

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

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(C) Given the position coordinates of the particle as $x = \alpha t^3$ and $y = \beta t^3$.The velocity components are obtained by differentiating the position with respect to time $t$:$v_x = \frac{dx}{dt} = \frac{d}{dt}(\alpha t^3) = 3\alpha t^2$$v_y = \frac{dy}{dt} = \frac{d}{dt}(\beta t^3) = 3\beta t^2$The speed $v$ of the particle is the magnitude of the velocity vector:$v = \sqrt{v_x^2 + v_y^2}$$v = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2}$$v = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4}$$v = \sqrt{9t^4(\alpha^2 + \beta^2)}$$v = 3t^2\sqrt{\alpha^2 + \beta^2}$

The coordinates of a moving particle at any time $t$ are given by $x = \alpha t^3$ and $y = \beta t^3$. The speed of the particle at time $t$ is given by

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