Question

The co-ordinates of a moving particle at any time ${}^{\prime }{t}^{\prime }$ are given by $x=\alpha t^3$ and $y=\beta t^3$. The speed of the particle at time ${}^{\prime }{t}^{\prime }$ is given by

• A. $3t\sqrt{{\alpha }^2+{\beta }^2}$
• B. $3t^2\sqrt{{\alpha }^2+{\beta }^2}$
• C. $t^2\sqrt{{\alpha }^2+{\beta }^2}$
• D. $\sqrt{{\alpha }^2+{\beta }^2}$

Answer 1

The correct option is B $3t^2\sqrt{{\alpha }^2+{\beta }^2}$

Coordinate of moving particle at time $t$ are $x=\alpha t^3$ and $y=\beta t^3.$

$v_x=\frac{dx}{dt}=3\alpha t^2\text{and}{v}_y=\frac{dy}{dt}=3\beta t^2$

$\therefore v=\sqrt{v_x^2+v_y^2}=\sqrt{9{\alpha }^2{t}^4+9{\beta }^2{t}^4}$

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

Hence, $\left(\text{B}\right)$ will be the correct option.