1. The position coordinates are given by x=lpha t^3, \qquad y=\beta t^3

2. Velocity components are the time derivatives of the coordinates: $v_x=\frac{dx}{dt}=3\alpha t^2$ $v_y=\frac{dy}{dt}=3\beta t^2$

3. The speed is the magnitude of the velocity vector: $v=\sqrt{v_x^2+v_y^2}$

4. Substitute the components: $v=\sqrt{(3\alpha t^2)^2+(3\beta t^2)^2}$ $=\sqrt{9\alpha^2 t^4+9\beta^2 t^4}$ $=\sqrt{9t^4(\alpha^2+\beta^2)}$ $=3t^2\sqrt{\alpha^2+\beta^2}$

5. Compare with the options:

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

Therefore, the correct answer is B.