1. Understanding the Problem

We have a classic collision problem with a few key elements:

* Ball 1: 30 g (0.03 kg) ball, moving horizontally with initial speed v0.

* Ball 2: 85 g (0.085 kg) ball, hanging motionless.

* Collision: Perfectly elastic, meaning kinetic energy is conserved.

* String: 1.2 m long, holding Ball 2.

2. Conservation of Momentum

In a perfectly elastic collision, both momentum and kinetic energy are conserved. Let's focus on momentum first:

* Before the collision: The total momentum is just the momentum of Ball 1:

p_initial = m1 * v0

* After the collision: The total momentum is the combined momentum of both balls:

p_final = m1 * v1 + m2 * v2

(where v1 and v2 are the final velocities of Ball 1 and Ball 2, respectively).

Since momentum is conserved, p_initial = p_final:

m1 * v0 = m1 * v1 + m2 * v2

3. Conservation of Kinetic Energy

Now, let's consider kinetic energy:

* Before the collision:

KE_initial = 1/2 * m1 * v0²

* After the collision:

KE_final = 1/2 * m1 * v1² + 1/2 * m2 * v2²

Since kinetic energy is conserved, KE_initial = KE_final:

1/2 * m1 * v0² = 1/2 * m1 * v1² + 1/2 * m2 * v2²

4. Solving for Final Velocities

We now have two equations and two unknowns (v1 and v2). Solving these equations simultaneously will give us the final velocities:

* Equation 1 (Momentum): m1 * v0 = m1 * v1 + m2 * v2

* Equation 2 (Kinetic Energy): 1/2 * m1 * v0² = 1/2 * m1 * v1² + 1/2 * m2 * v2²

The solution is:

* v1 = (m1 - m2) / (m1 + m2) * v0

* v2 = (2 * m1) / (m1 + m2) * v0

5. The Question

The prompt asks for the after which the ... It appears the question is incomplete. To continue, we need to know what you're looking for:

* What happens to the second ball? We can use the equation for v2 to find its final velocity and calculate how high it swings after the collision.

* What is the final velocity of the first ball? We can use the equation for v1 to find its final velocity.

Please provide the rest of the question so I can give you a complete answer!