Two masses each with mass 0.10kg are moving with velocities 3m/s along x axis and 5m/s along y-axis respectively. After an elastic collision one of the mass moves with a velocity `4 hati+4 hatj` . The energy of other mass after collision is `x/10` then x is

Explanation

To solve the problem step by step, we will use the principles of conservation of momentum and conservation of kinetic energy, as it is an elastic collision.

### Step 1: Identify the initial conditions
We have two masses, each with a mass \( m = 0.10 \, \text{kg} \). The first mass is moving with a velocity \( \vec{v_1} = 3 \hat{i} \, \text{m/s} \) and the second mass is moving with a velocity \( \vec{v_2} = 5 \hat{j} \, \text{m/s} \).

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (\( KE_i \)) of the system can be calculated using the formula:
\[ KE_i = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]
Substituting the values:
\[ KE_i = \frac{1}{2} (0.10) (3^2) + \frac{1}{2} (0.10) (5^2) \]
\[ = \frac{1}{2} (0.10) (9) + \frac{1}{2} (0.10) (25) \]
\[ = 0.05 \times 9 + 0.05 \times 25 = 0.45 + 1.25 = 1.70 \, \text{J} \]

### Step 3: Determine the final velocity of one mass
After the collision, one mass moves with a velocity \( \vec{v_1'} = 4 \hat{i} + 4 \hat{j} \). We need to calculate the kinetic energy of this mass.

### Step 4: Calculate the kinetic energy of the first mass after collision
The kinetic energy (\( KE_f \)) of the first mass after the collision is given by:
\[ KE_f = \frac{1}{2} m v_1'^2 \]
Calculating \( v_1' \):
\[ v_1' = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{m/s} \]
Now substituting this into the kinetic energy formula:
\[ KE_f = \frac{1}{2} (0.10) (4\sqrt{2})^2 \]
\[ = \frac{1}{2} (0.10) (32) = 0.05 \times 32 = 1.6 \, \text{J} \]

### Step 5: Apply conservation of kinetic energy
Since the collision is elastic, the total initial kinetic energy equals the total final kinetic energy:
\[ KE_i = KE_f + KE_{other} \]
Where \( KE_{other} \) is the kinetic energy of the second mass after the collision. Thus:
\[ 1.70 = 1.6 + KE_{other} \]
Solving for \( KE_{other} \):
\[ KE_{other} = 1.70 - 1.6 = 0.10 \, \text{J} \]

### Step 6: Relate \( KE_{other} \) to \( x \)
According to the problem, the energy of the other mass after the collision is given as \( \frac{x}{10} \). Therefore:
\[ 0.10 = \frac{x}{10} \]
Multiplying both sides by 10 gives:
\[ x = 1 \]

### Final Answer
Thus, the value of \( x \) is \( 1 \).

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