In a Broadway performance, an 85.0-kg actor swings from a R = 4.45-m-long cable that is horizontal when he starts. At the bottom of his arc, he picks up his 55.0-kg costar in an inelastic collision. What maximum height do they reach after their upward swing?
To find the maximum height reached by the actors, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the system is equal to the final mechanical energy.
1. The initial mechanical energy of the system is the sum of the potential energy and the kinetic energy of the actors at the bottom of their swing.
2. The potential energy at the bottom of the swing is given by the formula:
Potential Energy = m * g * h
Where:
- m is the combined mass of the actors (85.0 kg + 55.0 kg)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- h is the maximum height reached
3. The kinetic energy at the bottom of the swing is given by the formula:
Kinetic Energy = (1/2) * m * v^2
Where:
- m is the combined mass of the actors (85.0 kg + 55.0 kg)
- v is the velocity of the actors at the bottom of the swing
4. The final mechanical energy of the system is the potential energy of the actors at their maximum height. Since the actors are at rest at that height, their kinetic energy is zero.
Setting the initial mechanical energy equal to the final mechanical energy:
Potential Energy + Kinetic Energy = Potential Energy
m * g * h + (1/2) * m * v^2 = m * g * h
Canceling out the shared terms:
g * h + (1/2) * v^2 = g * h
Subtracting g * h from both sides of the equation:
(1/2) * v^2 = 0
Since v^2 = 0, v = 0. Therefore, the actors reach a maximum height of zero after their upward swing.
To find the maximum height reached after the swing, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system is conserved throughout the swing.
The initial mechanical energy of the system is given by:
E_initial = KE_actor_initial + KE_costar_initial + PE_initial
The final mechanical energy of the system at the maximum height is given by:
E_final = KE_actor_final + KE_costar_final + PE_final
Since the collision is inelastic, kinetic energy is not conserved after the collision. However, the potential energy at the maximum height is equal to zero.
Therefore, equating the initial and final mechanical energies, we have:
E_initial = E_final
KE_actor_initial + KE_costar_initial + PE_initial = KE_actor_final + KE_costar_final + PE_final
At the bottom of the swing, the actor is at his maximum speed and the costar is stationary. Therefore, the initial kinetic energy of the actor is given by:
KE_actor_initial = (1/2) * m_actor * v_actor_initial^2
where m_actor is the mass of the actor and v_actor_initial is the initial velocity of the actor.
The initial kinetic energy of the costar is zero:
KE_costar_initial = 0
The initial potential energy is given by:
PE_initial = m_actor * g * h_initial
where g is the acceleration due to gravity and h_initial is the initial height of the system.
At the maximum height, both the actor and the costar are momentarily stationary. Therefore, the final kinetic energies for both the actor and the costar are zero:
KE_actor_final = 0
KE_costar_final = 0
Considering that the final potential energy is zero:
PE_final = 0
Now we can equate the initial and final mechanical energies:
KE_actor_initial + KE_costar_initial + PE_initial = KE_actor_final + KE_costar_final + PE_final
Plugging in the given values:
(1/2) * m_actor * v_actor_initial^2 + 0 + m_actor * g * h_initial = 0 + 0 + 0
Simplifying the equation:
(1/2) * m_actor * v_actor_initial^2 + m_actor * g * h_initial = 0
Solving for h_initial, we have:
h_initial = - (1/2) * v_actor_initial^2 / g
Substituting the values:
h_initial = - (1/2) * (v_actor_initial)^2 / g
To find the initial velocity of the actor, we can use the conservation of mechanical energy:
E_initial = E_final
(1/2) * m_actor * v_actor_initial^2 + 0 + m_actor * g * R = 0 + 0 + 0
Simplifying the equation:
(1/2) * m_actor * v_actor_initial^2 + m_actor * g * R = 0
Solving for v_actor_initial, we have:
v_actor_initial = sqrt( - 2 * g * R)
Substituting this value into the equation for h_initial:
h_initial = - (1/2) * ((sqrt( - 2 * g * R))^2) / g
Simplifying further:
h_initial = - (1/2) * (2 * R)
h_initial = - R
Since height cannot be negative, the maximum height reached after the swing is:
h_max = |h_initial| = R = 4.45 m
Therefore, the actors reach a maximum height of 4.45 meters after their upward swing.