A particle of mass m having velocity v moving towards north collides with similar particle moving with same velocity towards east. The two particles stick together and move towards north east with a velocity

To find the velocity of two particles that collide and stick together, we can use the principles of conservation of momentum.

Let:
– Mass of each particle = m
– Initial velocity of the first particle (moving north) = v
– Initial velocity of the second particle (moving east) = v

1. Momentum Before Collision:
– Momentum of the first particle (north): p₁ = m * v
– Momentum of the second particle (east): p₂ = m * v

2. Total Momentum Before Collision:
– The momentum vector of the first particle is (0, mv) (north direction).
– The momentum vector of the second particle is (mv, 0) (east direction).
– Therefore, the total momentum vector before the collision is:
P_initial = (mv, mv)

3. Momentum After Collision:
– Since the two particles stick together after the collision, the combined mass is 2m.
– Let the velocity of the combined mass after the collision be V, and its direction will be towards the northeast.

4. Using Pythagoras’ Theorem:
– The magnitude of the momentum vector after the collision can be found using:
P_final = √[(mv)² + (mv)²] = √[2(mv)²] = mv√2

5. Calculating the Final Velocity:
– The total momentum after the collision is equal to the momentum before the collision:
P_final = 2m * V
– Setting them equal:
mv√2 = 2m * V
– Canceling m from both sides:
v√2 = 2V
– Solving for V:
V = (v√2) / 2 = v / √2

Final Answer:
The velocity of the combined mass after the collision is v/√2.