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)

$ 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}}} $