A 4.90-kg ball, moving to the right at a velocity of +2.50 m/s on a frictionless table, collides head-on with a stationary 8.00-kg ball. Find the final velocities of the balls if the collision meet the following conditions.

(a) elastic
4.9-kg ball =
8-kg ball =

(b) completely inelastic
Both =

(a) v₁= (m₁-m₂)v₁₀/(m₁+m₂),

v₂={2m₁v₁₀}/(m₁+m₂).
(b)
u=m₁V₁₀/(m₁+m₂)

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To find the final velocities of the balls in each case, we can apply the principles of conservation of momentum and conservation of kinetic energy.

(a) Elastic Collision:
In an elastic collision, both momentum and kinetic energy are conserved. The equation for the conservation of momentum in one dimension is:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

Given:
m1 (4.9 kg) and v1_initial (+2.5 m/s) for the first ball,
m2 (8.0 kg) and v2_initial (0 m/s) for the second ball.

Substituting the given values into the equation, we get:

(4.9 kg) * (2.5 m/s) + (8.0 kg) * (0 m/s) = (4.9 kg) * v1_final + (8.0 kg) * v2_final

Simplifying the equation, we have:

12.25 kg*m/s = 4.9 kg * v1_final + 0 kg * v2_final
12.25 kg*m/s = 4.9 kg * v1_final

Since v2_final is zero (the second ball is stationary after the collision), we can solve for v1_final:

v1_final = (12.25 kg*m/s) / (4.9 kg)
v1_final = 2.5 m/s

Therefore, the final velocity of the 4.9 kg ball is +2.5 m/s in the rightward direction, and the final velocity of the 8.0 kg ball is 0 m/s.

(b) Completely Inelastic Collision:
In a completely inelastic collision, the two objects stick together and move as one after the collision. Only momentum is conserved, while kinetic energy is not conserved.

Using the same equation for momentum conservation as in the elastic collision:

m1 * v1_initial + m2 * v2_initial = (m1 + m2) * v_final

Given the same values as in the elastic collision:

(4.9 kg) * (2.5 m/s) + (8.0 kg) * (0 m/s) = (4.9 kg + 8.0 kg) * v_final

Simplifying the equation:

12.25 kg*m/s = 12.9 kg * v_final

Solving for v_final:

v_final = (12.25 kg*m/s) / (12.9 kg)
v_final ≈ 0.949 m/s

Therefore, in a completely inelastic collision, the final velocity of both balls will be approximately 0.949 m/s.