A 4 kg mass and 1 kg mass are moving with equal linear momenta.The ratio of their kinetic energies is

Options

A. `1:2`
B. ` 2:1`
C. `1:4`
D. `4:1 `

Explanation

To find the ratio of the kinetic energies of two masses (4 kg and 1 kg) moving with equal linear momenta, we can follow these steps:

### Step 1: Understand the relationship between momentum and kinetic energy
The momentum \( p \) of an object is given by:
\[ p = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity.

The kinetic energy \( KE \) of an object is given by:
\[ KE = \frac{1}{2} m v^2 \]

### Step 2: Express velocity in terms of momentum
From the momentum equation, we can express velocity as:
\[ v = \frac{p}{m} \]

### Step 3: Substitute velocity into the kinetic energy formula
Substituting \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left( \frac{p}{m} \right)^2 = \frac{1}{2} m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m} \]

### Step 4: Write the kinetic energy for both masses
For mass \( m_1 = 4 \, \text{kg} \):
\[ KE_1 = \frac{p^2}{2 \cdot 4} = \frac{p^2}{8} \]

For mass \( m_2 = 1 \, \text{kg} \):
\[ KE_2 = \frac{p^2}{2 \cdot 1} = \frac{p^2}{2} \]

### Step 5: Find the ratio of kinetic energies
Now, we can find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{8}}{\frac{p^2}{2}} = \frac{p^2}{8} \cdot \frac{2}{p^2} = \frac{2}{8} = \frac{1}{4} \]

### Step 6: State the final result
Thus, the ratio of their kinetic energies is:
\[ KE_1 : KE_2 = 1 : 4 \]