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

Answer 1

Answer: (B)

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

Answer 2

$x=\alpha t^{3}, y=\beta t^{3}$
$v_{x}=\frac{d x}{d t}=3 \alpha t^{2}$
$v_{y}=\frac{d y}{d t}=3 \beta t^{2}$
Resultant velocity $v= \sqrt{v_{x}^{2}+v_{y}^{2}}$
$=\sqrt{9 \alpha^{2} t^{2}+9 \beta^{2} t^{4}}$
$=3 t^{2}=\sqrt{\alpha^{2}+\beta^{2}}$