Two bodies of equal masses move with uniform velocities `upsilon and 3 upsilon` respectively. Find the ratio of their kinetic energies.

Explanation

To find the ratio of the kinetic energies of two bodies with equal masses moving at different velocities, we can follow these steps:

### Step 1: Write the formula for kinetic energy
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

### Step 2: Define the kinetic energies of both bodies
Let the mass of both bodies be \( m \).
- For the first body moving with velocity \( v \):
\[ KE_1 = \frac{1}{2} m v^2 \]

- For the second body moving with velocity \( 3v \):
\[ KE_2 = \frac{1}{2} m (3v)^2 \]

### Step 3: Simplify the kinetic energy of the second body
Now, simplify \( KE_2 \):
\[ KE_2 = \frac{1}{2} m (3v)^2 = \frac{1}{2} m (9v^2) = \frac{9}{2} mv^2 \]

### Step 4: Find the ratio of the kinetic energies
Now, we can find the ratio of the kinetic energies \( KE_1 \) to \( KE_2 \):
\[ \text{Ratio} = \frac{KE_1}{KE_2} = \frac{\frac{1}{2} mv^2}{\frac{9}{2} mv^2} \]

### Step 5: Cancel out common terms
Since both kinetic energies have the same mass \( m \) and the common factor \( \frac{1}{2} \), we can cancel them out:
\[ \text{Ratio} = \frac{1}{9} \]

### Step 6: Write the final ratio
Thus, the ratio of the kinetic energies of the two bodies is:
\[ \text{Ratio} = 1 : 9 \]