In a Broadway performance an 81.0 kg actor swings from a 2.35 m long cable that is horizontal when he starts. At the bottom of his arc he picks up his 35.0 kg costar in an inelastic collision. (a) What is the velocity of the swinging actor just before picking up the costar?
To find the velocity of the swinging actor just before picking up the costar, we can use conservation of mechanical energy.
The initial potential energy of the actor is equal to the final kinetic energy of the system (actor + costar).
1. Calculate the potential energy of the actor at the bottom of the arc:
Potential energy (U) = mgh
Where m is the mass of the actor and costar combined, g is the acceleration due to gravity, and h is the height of the swing (which is not given, assuming 0.0 m for simplicity).
U = (81.0 kg + 35.0 kg) * 9.8 m/s^2 * 0.0 m
U = 0 J
2. Calculate the kinetic energy of the actor just before picking up the costar:
Kinetic energy (K) = 1/2 * m * v^2
Where m is the mass of the actor, costar, and the velocity just before picking up the costar, v is the velocity just before picking up the costar (which is what we are trying to find).
K = 1/2 * 81.0 kg * v^2
Since there is no change in height (potential energy) and no external work, the total mechanical energy of the system remains constant.
Therefore, the initial potential energy (0 J) is equal to the final kinetic energy (K):
0 J = 1/2 * 81.0 kg * v^2
3. Solve for the velocity (v):
0 J = 1/2 * 81.0 kg * v^2
v^2 = 0 J * 2 / 81.0 kg
v^2 = 0 J
v = 0 m/s
Hence, the velocity of the swinging actor just before picking up the costar is 0 m/s.
To find the velocity of the swinging actor just before picking up the costar, we can use the principle of conservation of mechanical energy. The swinging actor initially has gravitational potential energy since he is at a height, and as he swings down, this potential energy is converted into kinetic energy. At the bottom of his swing, all of the initial gravitational potential energy is converted into kinetic energy.
First, let's calculate the initial potential energy of the swinging actor:
Potential energy (PE) = mass x gravity x height
PE = 81.0 kg x 9.8 m/s^2 x 2.35 m
Next, let's calculate the final kinetic energy of the swinging actor:
Kinetic energy (KE) = (1/2) x mass x velocity^2
KE = (1/2) x 81.0 kg x velocity^2
According to the conservation of mechanical energy, the initial potential energy equals the final kinetic energy:
PE = KE
Setting these two equations equal to each other:
81.0 kg x 9.8 m/s^2 x 2.35 m = (1/2) x 81.0 kg x velocity^2
Now, we can solve for the velocity of the swinging actor just before picking up the costar:
velocity^2 = (2 x 81.0 kg x 9.8 m/s^2 x 2.35 m) / 81.0 kg
velocity^2 = 2 x 9.8 m/s^2 x 2.35 m
velocity^2 = 45.95 m^2/s^2
Taking the square root of both sides to solve for velocity:
velocity = sqrt(45.95 m^2/s^2)
The velocity of the swinging actor just before picking up the costar is approximately 6.77 m/s.