Question

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

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

$x=\alpha \,\,{{t}^{3}},\,\,y=\beta {{t}^{3}}$ ${{v}_{x}}=\frac{dx}{dt}=3\alpha {{t}^{2}}$ ${{v}_{y}}=\frac{dy}{dt}=3\beta {{t}^{2}}$ Resultant velocity, $v=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}$ $=\sqrt{9{{\alpha }^{2}}{{t}^{4}}+9{{\beta }^{2}}{{t}^{4}}}$ $=3{{t}^{2}}\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}}$