The coordinates of a moving particle at any t | vedclass

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(D) The position vector of the particle is given by $\vec{r} = x \hat{i} + y \hat{j} = \alpha t^3 \hat{i} + \beta t^3 \hat{j}$.To find the velocity ve

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

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(D) The position vector of the particle is given by $\vec{r} = x \hat{i} + y \hat{j} = \alpha t^3 \hat{i} + \beta t^3 \hat{j}$.To find the velocity vector $\vec{v}$,we differentiate the position vector with respect to time $t$:$\vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(\alpha t^3 \hat{i} + \beta t^3 \hat{j}) = 3\alpha t^2 \hat{i} + 3\beta t^2 \hat{j}$.The speed is the magnitude of the velocity vector:$v = |\vec{v}| = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2}$.$v = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4}$.$v = \sqrt{9 t^4 (\alpha^2 + \beta^2)}$.$v = 3 t^2 \sqrt{\alpha^2 + \beta^2}$.

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

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