a ball of mass 4kg moving with a velocity of 12m/s impinges directly an another ball of mass 8kg moving with a velocity of 4m/s in same direction.find their velocities after impact.

To solve the problem of two colliding balls, we can use the principle of conservation of momentum. This principle states that the total momentum of a closed system before a collision is equal to the total momentum after the collision, provided no external forces act on it. Let's break this down step by step.

Understanding Momentum

Momentum is defined as the product of an object's mass and its velocity. Mathematically, it can be expressed as:

Momentum (p) = Mass (m) × Velocity (v)

Calculating Initial Momentum

In this scenario, we have two balls:

• Ball 1: Mass = 4 kg, Velocity = 12 m/s
• Ball 2: Mass = 8 kg, Velocity = 4 m/s

Now, let's calculate the initial momentum of each ball:

• Momentum of Ball 1: p1 = m1 × v1 = 4 kg × 12 m/s = 48 kg·m/s
• Momentum of Ball 2: p2 = m2 × v2 = 8 kg × 4 m/s = 32 kg·m/s

The total initial momentum (P_initial) of the system is:

P_initial = p1 + p2 = 48 kg·m/s + 32 kg·m/s = 80 kg·m/s

Applying Conservation of Momentum

After the collision, let’s denote the final velocities of Ball 1 and Ball 2 as v1' and v2', respectively. According to the conservation of momentum:

P_initial = P_final

This can be expressed as:

m1 × v1 + m2 × v2 = m1 × v1' + m2 × v2'

Substituting the known values:

4 kg × 12 m/s + 8 kg × 4 m/s = 4 kg × v1' + 8 kg × v2'

Which simplifies to:

80 kg·m/s = 4v1' + 8v2'

Assuming Perfectly Inelastic Collision

If we assume that the balls stick together after the collision (a perfectly inelastic collision), they will have a common final velocity (v'). In this case, we can express the equation as:

P_initial = (m1 + m2) × v'

Substituting the total mass and initial momentum:

80 kg·m/s = (4 kg + 8 kg) × v'

80 kg·m/s = 12 kg × v'

Now, solving for v':

v' = 80 kg·m/s / 12 kg = 6.67 m/s

Final Velocities

Thus, after the collision, if we assume a perfectly inelastic collision, both balls will move together with a final velocity of approximately 6.67 m/s.

In summary, by applying the conservation of momentum, we determined that the final velocity of the two balls after colliding and sticking together is about 6.67 m/s. This approach illustrates how fundamental physics principles can be used to analyze real-world scenarios involving motion and collisions.