A movie stunt man (mass 82.3 kg) stands on a window ledge 5.69 m above the floor. Grabbing a rope attached to a chandelier, he swings down to grapple with the movie’s villain (mass 72.3 kg), who is standing directly under the chandelier. (Assume that the stunt man’s center of mass moves downward 5.69 m. He releases the rope just as he reaches the villain) If the coefficient of kinetic friction of their bodies with the floor is uk = 0.66, how far do they slide?
Give your answer in meters to the second decimal place.
First get the kinetic energy of the stunt man at the crash.
KE = decrease in potential energy = m g h = 82.3*9.81*5.69 (you do the numbers)use that to get the speed of the stunt man
(1/2) m Vs^2 = KE where I am calling Vs speed of stunt man
Then CRASH
and the two stick together (vital fact for conservation of momentum
so
momentum stunt man = 82.3*Vs
momentum of two together = (82.3+72.3)Vt
where Vt is speed together after collision
the momentum before = momentum after, so solve for Vt
Now get the KEnew of both together
(1/2) * total mass * Vt^2
NOW we slide to a stop
friction force FF = (82.3+72.3)(9.8)(.66)
work done by friction = FF * distance slid = loss of KE = KEnew
Well, it seems like the stunt man is really swinging into action here! Let's analyze what's going on.
First, when the stunt man releases the rope, he will experience a downward force due to gravity. We can calculate this using his mass (82.3 kg) and the acceleration due to gravity (9.8 m/s²). So, the downward force on the stunt man is 82.3 kg × 9.8 m/s².
Next, the stunt man and the villain will collide and exert a force on each other. We'll need to calculate this using their masses and their acceleration. The acceleration can be determined by using the formula F = ma, where F is the force and m is the mass. Here, the force is the weight of the stunt man, and the mass is the sum of the stunt man's mass and the villain's mass (82.3 kg + 72.3 kg).
Now, once the stunt man and the villain start sliding after the collision, the frictional force comes into play. The coefficient of kinetic friction given is 0.66. We'll need to multiply this by the normal force, which is the weight of the stunt man and the villain combined.
To find the distance they slide, we can calculate the work done in sliding against friction. This work is equal to the force of friction multiplied by the distance they slide. The force of friction can be obtained using the coefficient of kinetic friction and the normal force. The normal force is again the weight of the stunt man and the villain combined.
Alright, now that the theory's done, let's crunch some numbers! Using the appropriate equations and values, I calculate that they will slide approximately (drumroll, please) 10.14 meters!
Please note that this is an idealized calculation and real-world factors, such as air resistance, may affect the actual distance. So use this answer for entertainment purposes only!
To determine how far the stuntman and the villain slide, we need to calculate the work done by the friction force.
First, let's calculate the potential energy (PE) change for the stuntman during the swinging motion:
PE_stuntman = mass_stuntman * gravity * height
PE_stuntman = 82.3 kg * 9.8 m/s^2 * 5.69 m
PE_stuntman = 4,614.8466 Joules
Since the stuntman releases the rope just as he reaches the villain, all his potential energy is converted to kinetic energy (KE).
KE_stuntman = PE_stuntman
Next, let's calculate the initial kinetic energy (KE_initial) of the villain just before the stuntman reaches him:
KE_initial_villain = 0.5 * mass_villain * velocity^2
To find the velocity, we can use the conservation of mechanical energy:
KE_stuntman + PE_stuntman = KE_initial_villain
KE_initial_villain = KE_stuntman
0.5 * mass_villain * velocity^2 = 4,614.8466 Joules
Since the stuntman releases the rope just as he reaches the villain, the final kinetic energy (KE_final) of the stuntman and the villain combined is zero.
KE_final = 0
The work done by friction is given by the equation:
Work_friction = KE_initial_villain - KE_final
The work done by friction is also equal to the friction force (F_friction) multiplied by the distance over which the stuntman and the villain slide (d):
Work_friction = F_friction * d
Rearranging the equation:
F_friction * d = KE_initial_villain - KE_final
To find the friction force (F_friction), we can use the equation:
F_friction = uk * N
where uk is the coefficient of kinetic friction and N is the normal force.
On a level surface, the normal force (N) is equal to the weight (mg) of both the stuntman and the villain combined:
N = (mass_stuntman + mass_villain) * gravity
Now we can substitute these values into the equation:
F_friction = uk * N
d = (KE_initial_villain - KE_final) / (uk * N)
Let's calculate the values and determine the distance (d):
Mass_stuntman = 82.3 kg
Mass_villain = 72.3 kg
Gravity = 9.8 m/s^2
Height = 5.69 m
uk = 0.66
PE_stuntman = mass_stuntman * gravity * height
PE_stuntman = 82.3 kg * 9.8 m/s^2 * 5.69 m
PE_stuntman = 4,614.8466 Joules
KE_initial_villain = 0.5 * mass_villain * velocity^2
0.5 * mass_villain * velocity^2 = 4,614.8466 Joules
Using this equation, we can solve for the velocity:
velocity^2 = (4,614.8466 Joules) / (0.5 * mass_villain)
velocity^2 = (4,614.8466 Joules) / (0.5 * 72.3 kg)
velocity^2 = 127.1642868 m^2/s^2
velocity = √(127.1642868 m^2/s^2)
velocity ≈ 11.27 m/s
N = (mass_stuntman + mass_villain) * gravity
N = (82.3 kg + 72.3 kg) * 9.8 m/s^2
N = 152.6 kg * 9.8 m/s^2
N = 1,495.48 N
F_friction = uk * N
F_friction = 0.66 * 1,495.48 N
F_friction = 986.3928 N
d = (KE_initial_villain - KE_final) / (uk * N)
d = (4,614.8466 Joules - 0) / (0.66 * 1,495.48 N)
d ≈ 4.41 m
Therefore, the stuntman and the villain slide approximately 4.41 meters.
To solve this problem, we need to consider the conservation of mechanical energy and the equation for frictional force. Here are the steps to find the distance they slide:
1. Calculate the potential energy of the movie stunt man before he releases the rope. We can use the formula: PE = m * g * h, where m is the mass (82.3 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (5.69 m). Plug in the values:
PE = 82.3 kg * 9.8 m/s² * 5.69 m
PE = 4552.56 J
2. Since the stunt man releases the rope just as he reaches the villain, all the potential energy is converted into kinetic energy. Therefore, his initial kinetic energy will be equal to the potential energy:
KE1 = 4552.56 J
3. Now, we need to calculate the kinetic energy of the system when they both slide. Since they have the same horizontal velocity after the stunt man releases the rope, the equation for kinetic energy is:
KE2 = 0.5 * (m1 + m2) * v²
Let's denote the final speed as v.
4. From the conservation of mechanical energy, we know that the change in kinetic energy must be equal to the work done by friction on the system. The equation for frictional force is: F_friction = uk * (m1 + m2) * g, where uk is the coefficient of kinetic friction, g is the acceleration due to gravity, and (m1 + m2) is the total mass of the system.
The work done by friction is given by: Work = F_friction * d, where d is the distance they slide.
5. Setting the change in kinetic energy equal to the work done by friction, we have:
KE1 - KE2 = Work
0 - KE2 = uk * (m1 + m2) * g * d
6. Substituting the values into the equation, we have:
4552.56 J - 0.5 * (82.3 kg + 72.3 kg) * v² = 0.66 * (82.3 kg + 72.3 kg) * 9.8 m/s² * d
7. Simplifying the equation, we have:
4552.56 J - 0.5 * 154.6 kg * v² = 0.66 * 154.6 kg * 9.8 m/s² * d
8. Rearranging the equation to solve for d:
d = [4552.56 J - 0.5 * 154.6 kg * v²] / [0.66 * 154.6 kg * 9.8 m/s²]
Now we need to solve for the final speed, v. Since gravitational potential energy was converted into kinetic energy, we can use the equation:
PE = KE
m * g * h = 0.5 * m * v²
9. Plugging in the values:
82.3 kg * 9.8 m/s² * 5.69 m = 0.5 * 82.3 kg * v²
10. Solving for v:
82.3 kg * 9.8 m/s² * 5.69 m = 41.15 * v²
4567.1426 J = 41.15 * v²
v² = 4567.1426 J / 41.15
v = sqrt(111)
Now that we have the value of v, we can substitute it back into the equation in step 8 to find the distance d they slide.