Question

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 :

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

Answer 1

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

$\text{Step 1:}$$\text{Calculate velocity in x and y direction}$

Velocity in $x$ direction:

$V_x=\frac{dx}{dt}$ $=\frac{d}{dt}\left(\alpha t^3\right)$ $=3\alpha t^2$

Velocity in $y$ direction:

$V_y=\frac{dy}{dt}$ $=\frac{d}{dt}\left(\beta t^3\right)$$=3\beta t^2$

$\text{Step 2: Calculate speed}$

$V=\sqrt{V_x^2+V_y^2}$$=\sqrt{(3\alpha t^2{)}^2+(3\beta t^2{)}^2}$

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

Hence option $C$ is correct