The coordinates of a moving particle at any time't' are given by `x = alphat^(3)` and `y = betat^(3)`. The speed of the particle at time 't' is given by

Options

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))`

Answer

To find the speed of the particle at any time 't', we will follow these steps:

### Step 1: Differentiate the position functions
The coordinates of the particle are given as:
- \( x = \alpha t^3 \)
- \( y = \beta t^3 \)

We need to find the velocities in both the x and y directions by differentiating these equations with respect to time \( t \).

### Step 2: Find the velocity in the x-direction
To find the velocity in the x-direction (\( v_x \)), we differentiate \( x \) with respect to \( t \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt} (\alpha t^3) = \alpha \cdot \frac{d}{dt}(t^3) = \alpha \cdot 3t^2 = 3\alpha t^2 \]

### Step 3: Find the velocity in the y-direction
Next, we find the velocity in the y-direction (\( v_y \)) by differentiating \( y \) with respect to \( t \):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt} (\beta t^3) = \beta \cdot \frac{d}{dt}(t^3) = \beta \cdot 3t^2 = 3\beta t^2 \]

### Step 4: Calculate the speed of the particle
The speed of the particle is the magnitude of the velocity vector, which can be calculated using the Pythagorean theorem:
\[ \text{Speed} = v = \sqrt{v_x^2 + v_y^2} \]

Substituting the expressions for \( v_x \) and \( v_y \):
\[ v = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2} \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ v = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4} = \sqrt{9t^4(\alpha^2 + \beta^2)} = 3t^2 \sqrt{\alpha^2 + \beta^2} \]

### Final Result
Thus, the speed of the particle at time \( t \) is given by:
\[ \text{Speed} = 3t^2 \sqrt{\alpha^2 + \beta^2} \]

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