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

Options

A. (a)` sqrt ( alpha ^ 2 + beta ^ 2 ) `
B. (b)`3t sqrt ( alpha ^ 2 + beta ^2 ) `
C. (c)`3t^ 2 sqrt ( alpha ^ 2 + beta ^ 2 ) `
D. (d)`t ^ 2 sqrt ( alpha ^ 2 + beta ^ 2 ) `

Explanation

To find the speed of the particle at time \( t \), we start with the given coordinates:

1. **Given Coordinates**:
\[ x = \alpha t^3 \]
\[ y = \beta t^3 \]

2. **Calculate the Velocity Components**:
- The velocity in the x-direction (\( v_x \)) is the derivative of \( x \) with respect to \( t \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(\alpha t^3) = 3\alpha t^2 \]
- The velocity in the y-direction (\( v_y \)) is the derivative of \( y \) with respect to \( t \):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(\beta t^3) = 3\beta t^2 \]

3. **Calculate the Speed**:
- The speed \( v \) of the particle is given by the magnitude of the velocity vector, which can be calculated using the Pythagorean theorem:
\[ v = \sqrt{v_x^2 + v_y^2} \]
- Substituting the expressions for \( v_x \) and \( v_y \):
\[ v = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2} \]
- Simplifying this expression:
\[ v = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4} \]
\[ v = \sqrt{9t^4(\alpha^2 + \beta^2)} \]
- Taking the square root:
\[ v = 3t^2\sqrt{\alpha^2 + \beta^2} \]

4. **Final Expression**:
- Thus, the speed of the particle at time \( t \) is:
\[ v = 3t^2\sqrt{\alpha^2 + \beta^2} \]