A particle of mass 10 kg moving eastwards with a speed 5 `ms^(-1)` collides with another particle of the same mass moving north-wards with the same speed 5`ms^(-1)` .The two particles coalesce on collision.The new particle of mass 20 kg will move in the north-east direction with velocity

Options

A. `10 ms^(-1)`
B. `5 ms^(-1)`
C. `(5//sqrt(2))ms^(-1)`
D. None of these

Explanation

To solve the problem, we need to determine the velocity of the new particle formed after the collision of the two particles. Let's break down the steps:

### Step 1: Identify the initial conditions
- Mass of particle 1 (m1) = 10 kg, moving east with speed (v1) = 5 m/s.
- Mass of particle 2 (m2) = 10 kg, moving north with speed (v2) = 5 m/s.

### Step 2: Determine the momentum of each particle
- Momentum of particle 1 (P1) = m1 * v1 = 10 kg * 5 m/s = 50 kg·m/s (east direction).
- Momentum of particle 2 (P2) = m2 * v2 = 10 kg * 5 m/s = 50 kg·m/s (north direction).

### Step 3: Calculate the total momentum before collision
Since the two particles are moving at right angles to each other, we can treat their momenta as vectors:
- Total momentum in the east direction (x-direction) = P1 = 50 kg·m/s.
- Total momentum in the north direction (y-direction) = P2 = 50 kg·m/s.

### Step 4: Combine the momenta to find the resultant momentum
The total momentum vector (P_total) can be found using the Pythagorean theorem:
\[ P_{\text{total}} = \sqrt{(P1)^2 + (P2)^2} = \sqrt{(50)^2 + (50)^2} = \sqrt{2500 + 2500} = \sqrt{5000} = 50\sqrt{2} \text{ kg·m/s}. \]

### Step 5: Determine the mass of the new particle
After the collision, the two particles coalesce, so the total mass (M) of the new particle is:
\[ M = m1 + m2 = 10 \text{ kg} + 10 \text{ kg} = 20 \text{ kg}. \]

### Step 6: Calculate the velocity of the new particle
Using the conservation of momentum, the velocity (v) of the new particle can be calculated as:
\[ P_{\text{total}} = M \cdot v \]
\[ 50\sqrt{2} = 20 \cdot v \]
\[ v = \frac{50\sqrt{2}}{20} = \frac{5\sqrt{2}}{2} \text{ m/s}. \]

### Step 7: Determine the direction of the new particle
The new particle moves in the northeast direction, which is at a 45-degree angle to both the east and north directions.

### Final Answer
The new particle will move in the northeast direction with a velocity of \( \frac{5\sqrt{2}}{2} \text{ m/s} \).

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