Q. 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) $3t\sqrt{\alpha^{2} + \ \beta^{2}}$
B) $3t^{2}\sqrt{\alpha^{2} + \ \beta^{2}}$
C) $t^{2}\sqrt{\alpha^{2} + \ \beta^{2}}$
D) $\sqrt{\alpha^{2} + \ \beta^{2}}$
Explanation
(b) $x = \alpha t^{3},\ y = \beta t^{3}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v_{x} = \frac{dx}{dt} = 3\alpha t^{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v_{y} = \frac{dy}{dt} = 3\ \beta t^{2}$
Resultant velocity, $v = \sqrt{v_{x}^{2} + v_{y}^{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \sqrt{9\alpha^{2}t^{4} + 9\beta^{2}t^{4}} = 3t^{2}\sqrt{\alpha^{2} + \ \beta^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.