In two systems of relations among velocity , acceleration , and force are , respectively , `v_(2) = (alpha^(2))/( beta) v_(1) , a_(2) = alpha beta a_(1), and F_(2) = (F_(1))/( alpha beta)`. If `alpha and beta` are constants , then make relations among mass , length , and time in two systems.

Explanation

To find the relations among mass, length, and time in two systems based on the given equations for velocity, acceleration, and force, we can follow these steps:

### Step 1: Relation between Mass
We start with the fundamental relation between force, mass, and acceleration:
\[ F = m \cdot a \]

From the given relations, we have:
1. \( F_2 = \frac{F_1}{\alpha \beta} \)
2. \( a_2 = \alpha \beta a_1 \)

Using the relation \( F = m \cdot a \), we can express the masses in terms of force and acceleration:

\[ \frac{F_2}{a_2} = \frac{F_1}{a_1} \]

Substituting the expressions for \( F_2 \) and \( a_2 \):

\[ \frac{\frac{F_1}{\alpha \beta}}{\alpha \beta a_1} = \frac{F_1}{a_1} \]

This simplifies to:

\[ \frac{F_1}{\alpha \beta \cdot a_1} = \frac{F_1}{a_1} \]

Cross-multiplying gives:

\[ F_1 \cdot a_1 = F_1 \cdot \alpha \beta \cdot a_1 \implies m_2 = \frac{m_1}{\alpha^2 \beta} \]

Thus, the relation between the masses is:

\[ m_2 = \frac{m_1}{\alpha^2 \beta} \]

### Step 2: Relation between Length
Next, we find the relation for length using the definitions of velocity and acceleration.

The dimensions of velocity \( v \) and acceleration \( a \) are:
- \( [v] = \frac{L}{T} \)
- \( [a] = \frac{L}{T^2} \)

Now, we can express the relation between lengths using the given velocity and acceleration relations.

From the velocity relation:
\[ v_2 = \frac{\alpha^2}{\beta} v_1 \]

Squaring both sides gives:
\[ v_2^2 = \left(\frac{\alpha^2}{\beta}\right)^2 v_1^2 = \frac{\alpha^4}{\beta^2} v_1^2 \]

For acceleration:
\[ a_2 = \alpha \beta a_1 \]

Now, substituting into the relation for length:
\[ \frac{v_2^2}{a_2} = \frac{\frac{\alpha^4}{\beta^2} v_1^2}{\alpha \beta a_1} \]

This simplifies to:
\[ \frac{v_2^2}{a_2} = \frac{\alpha^3}{\beta^3} \cdot \frac{v_1^2}{a_1} \]

Thus, the relation between lengths is:
\[ L_2 = \frac{\alpha^3}{\beta^3} L_1 \]

### Step 3: Relation between Time
Finally, we find the relation for time. The time can be expressed as:
\[ T = \frac{V}{A} \]

Using the relations:
\[ T_2 = \frac{v_2}{a_2} \]

Substituting the expressions for \( v_2 \) and \( a_2 \):
\[ T_2 = \frac{\frac{\alpha^2}{\beta} v_1}{\alpha \beta a_1} \]

This simplifies to:
\[ T_2 = \frac{\alpha}{\beta^2} T_1 \]

### Final Relations
Thus, the final relations among mass, length, and time in the two systems are:
1. \( m_2 = \frac{m_1}{\alpha^2 \beta} \)
2. \( L_2 = \frac{\alpha^3}{\beta^3} L_1 \)
3. \( T_2 = \frac{\alpha}{\beta^2} T_1 \)