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?
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To find the solutions to the given problem, we can apply the principles of conservation of energy and momentum.
(a) To find the velocity of the swinging actor just before picking up the costar, we can start by calculating the potential energy of the actor at the bottom of his arc.
Potential energy (PE) = m * g * h
where:
m = mass of the actor = 80.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the swing = half the length of the cable = 2.05 m / 2 = 1.025 m
PE = 80.0 kg * 9.8 m/s^2 * 1.025 m
PE = 799.4 J
At the bottom of the swing, all the potential energy is converted to kinetic energy (KE) since there is no horizontal movement.
KE = 799.4 J
Using the kinetic energy equation:
KE = 1/2 * m * v^2
where:
m = mass of the actor = 80.0 kg
v = velocity of the actor just before picking up the costar
799.4 J = 0.5 * 80.0 kg * v^2
799.4 J = 40.0 kg * v^2
Divide both sides of the equation by 40.0 kg:
v^2 = 799.4 J / 40.0 kg
v^2 = 19.985 m^2/s^2
Take the square root of both sides of the equation to find the velocity:
v = √(19.985 m^2/s^2)
v ≈ 4.47 m/s
Therefore, the velocity of the swinging actor just before picking up the costar is approximately 4.47 m/s.
(b) To find the velocity of both actors just after picking up the costar, we can use the principle of conservation of momentum.
Initial momentum of the swinging actor = final momentum of the swinging actor + final momentum of the costar
Before picking up the costar, only the swinging actor is in motion, so the initial momentum is:
Initial momentum = m1 * v1
where:
m1 = mass of the swinging actor = 80.0 kg
v1 = velocity of the swinging actor just before picking up the costar = 4.47 m/s
Initial momentum = 80.0 kg * 4.47 m/s = 357.6 kg·m/s
After picking up the costar, the mass of the system becomes the combined mass of the swinging actor and the costar (80.0 kg + 47.0 kg = 127.0 kg). Let's assume their combined velocity after the collision is v2.
Final momentum of the swinging actor and costar = (m1 + m2) * v2
where:
m2 = mass of the costar = 47.0 kg
Final momentum of the swinging actor and costar = 127.0 kg * v2
Using the principle of conservation of momentum:
Initial momentum = Final momentum
357.6 kg·m/s = 127.0 kg * v2
Divide both sides of the equation by 127.0 kg:
v2 = 357.6 kg·m/s / 127.0 kg
v2 ≈ 2.817 m/s
Therefore, the velocity of both actors just after picking up the costar is approximately 2.817 m/s.
(c) To find the maximum height they reach after their upward swing, we can use the conservation of energy principle.
Initially, the actors have kinetic energy (KE) and no potential energy (PE). At the top of their swing, their potential energy is maximum and their kinetic energy is zero.
Using the principle of conservation of energy:
Initial KE = Final PE
1/2 * (m1 + m2) * v2^2 = (m1 + m2) * g * h
where:
m1 = mass of the swinging actor = 80.0 kg
m2 = mass of the costar = 47.0 kg
v2 = velocity of both actors just after picking up the costar = 2.817 m/s
g = acceleration due to gravity = 9.8 m/s^2
h = maximum height they reach after their upward swing (to be solved)
Rearranging the equation:
h = v2^2 / (2 * g)
h = (2.817 m/s)^2 / (2 * 9.8 m/s^2)
h = 7.885 m^2/s^2 / 19.6 m/s^2
h ≈ 0.402 m
Therefore, the maximum height they reach after their upward swing is approximately 0.402 m.