In a Broadway performance an 80.0 kg actor swings from a 2.05 m long cable that is horizontal when he starts. At the bottom of his arc he picks up his 47.0 kg costar in an inelastic collision. (a) What is the velocity of the swinging actor just before picking up the costar? (b) What is the velocity of both actors just after picking up the costar? (c) What maximum height do they reach after their upward swing?

To solve this problem, we can use the principles of conservation of energy and momentum. Let's break down each part of the question:

(a) To find the velocity of the swinging actor just before picking up the costar, we need to consider the conservation of energy. The initial gravitational potential energy of the actor is transferred into kinetic energy as he swings. At the lowest point of his swing, his gravitational potential energy will be zero, so all the energy will be in the form of kinetic energy.

Mathematically, we can express this as:

m_actor * g * h_actor = (1/2) * m_actor * v_actor^2

Here,
m_actor = mass of the actor = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h_actor = length of the cable = 2.05 m
v_actor = velocity of the actor just before picking up the costar

We can rearrange the equation to solve for v_actor:

v_actor = sqrt(2 * g * h_actor)

Plugging in the given values, we can calculate the answer.

(b) To find the velocity of both actors just after picking up the costar, we need to consider the conservation of momentum. Momentum is conserved in an inelastic collision, so the total momentum of the actors before the collision will be equal to the total momentum after the collision.

Mathematically, we can express this as:

(m_actor * v_actor) + (m_costar * 0) = (m_actor + m_costar) * v_final

Here,
m_costar = mass of the costar = 47.0 kg
v_final = velocity of both actors just after picking up the costar

Since the costar is picked up from rest (0 velocity), we can simplify the equation to:

m_actor * v_actor = (m_actor + m_costar) * v_final

We can rearrange the equation to solve for v_final:

v_final = (m_actor * v_actor) / (m_actor + m_costar)

Plugging in the given values, we can calculate the answer.

(c) To find the maximum height reached after their upward swing, we need to use the conservation of mechanical energy. At the highest point of the swing, all the initial kinetic energy will be converted into gravitational potential energy.

Mathematically, we can express this as:

(1/2) * (m_actor + m_costar) * v_final^2 = (m_actor + m_costar) * g * h_max

Here,
h_max = maximum height reached after the swing

We can rearrange the equation to solve for h_max:

h_max = (v_final^2) / (2 * g)

Plugging in the value of v_final and the given acceleration due to gravity, we can calculate the answer.

So, by following these equations and plugging in the appropriate values, you can find the answers to all three parts of the question.