Problem 85 - University Physics with Modern Physics

Question

A movie stuntman (mass 80.0 \(\mathrm{kg}\) ) stands on a window ledge 5.0 \(\mathrm{m}\) above the floor (Fig. P8.85). Grabbing a rope attached to a chandelier, he swings down to grapple with the movie's villain (mass 70.0 kg), who is standing directly under the chandelier. (Assume that the stuntman's center of mass moves downward 5.0 \(\mathrm{m} .\) He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is \(\mu_{k}=0.250,\) how far do they slide?

Step-by-Step Solution

Answer

(a) The speed of the entwined foes is 3.49 m/s. (b) They slide 4.98 m across the floor.

Step 1: Determine the Initial Potential Energy

The stuntman initially stands on a ledge 5.0 m above the floor. To find his initial potential energy (PE), use the equation \( PE = mgh \), where \( m = 80.0 \text{ kg} \) (mass of the stuntman), \( g = 9.81 \text{ m/s}^2 \) (acceleration due to gravity), and \( h = 5.0 \text{ m} \) (height). Calculate \( PE = 80.0 \cdot 9.81 \cdot 5.0 \).

Step 2: Calculate the Initial Kinetic Energy

Initially, the stuntman is at rest, so his initial kinetic energy (KE) is \( 0 \text{ J} \).

Step 3: Use Conservation of Mechanical Energy

Since there is no air resistance or other forces doing work, the mechanical energy is conserved. The initial potential energy becomes the kinetic energy at the bottom. Set the initial PE equal to the final KE: \( mgh = \frac{1}{2} mv^2 \). Solve for \( v \) to find the speed just before reaching the villain.

Step 4: Consider the Collision

The collision between the stuntman and the villain is perfectly inelastic. Use the conservation of momentum to find the combined speed \( v_f \). The equation for momentum conservation is \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \), where \( m_1 = 80.0 \text{ kg} \) (stuntman), \( m_2 = 70.0 \text{ kg} \) (villain), and \( v_1 = v \) (from Step 3), \( v_2 = 0 \text{ m/s} \) (villain's initial speed). Solve for \( v_f \).

Step 5: Calculate the Frictional Force

To find how far they slide, calculate the frictional force using \( F_{friction} = \mu_k (m_1 + m_2)g \), where \( \mu_k = 0.250 \).

Step 6: Use Work-Energy Principle to Find Distance

The work done by friction (which is equal to the kinetic energy lost) is \( W = F_{friction} \times d = \frac{1}{2}(m_1 + m_2)v_f^2 \), where \( d \) is the distance they slide. Solve for \( d \) to find how far they slide.

Key Concepts

Kinetic Energy

When discussing kinetic energy, it's important to understand it's the energy an object possesses due to its motion. In the problem of the stuntman, once he releases the rope and begins to swing downwards, his potential energy is converted into kinetic energy. Kinetic energy in this context is calculated using the formula:

• \[KE = \frac{1}{2}mv^2\]where \( m \) is the mass and \( v \) is the velocity.

Initially, while the stuntman is standing still, his kinetic energy is zero, because he is not moving. However, as he swings down and reaches the greatest speed at the lowest point, his kinetic energy is at its maximum. This energy conversion illustrates the concept of conservation of energy, where potential energy is transformed into kinetic energy without loss due to external forces like air resistance in this ideal scenario.

Potential Energy

Potential energy is the stored energy of position possessed by an object. For the stuntman standing on the ledge 5 meters above the floor, potential energy is a key concept. This is calculated using the equation:

• \[PE = mgh\]where:
  • \( m \) is the mass of the stuntman (80.0 kg)
  • \( g \) is the acceleration due to gravity (approximately 9.81 \( m/s^2 \))
  • \( h \) is the height above the ground

Potential energy is at its peak when the stuntman is on the ledge. As he descends, this stored energy begins to convert into kinetic energy as he loses height. The transformation of potential into kinetic energy is crucial because it demonstrates how energy is conserved and transformed into different types through motion.

Inelastic Collision

An inelastic collision is one where two objects collide and stick together, resulting in a loss of kinetic energy, unlike elastic collisions where objects bounce off each other without loss of speed or energy. In the exercise, when the stuntman and villain come together, they constitute an inelastic collision. After collision, they travel together as one unit.To calculate the result of this inelastic collision, we use the conservation of momentum principle, expressed as:

• \[m_1v_1 + m_2v_2 = (m_1 + m_2)v_f\]

This equation allows us to find the velocity \( v_f \) of the combined mass after they have collided. Since both are moving together across the floor, the velocity \( v_f \) serves to show how kinetic energy is redistributed in an inelastic collision scenario.

Frictional Force

Frictional force is the force resisting the motion of surfaces sliding against each other. In this exercise, once the stuntman and villain collide and begin sliding across the floor, they are subjected to a frictional force. The frictional force is calculated by:

• \[F_{friction} = \mu_k (m_1 + m_2)g\]where:
  • \( \mu_k \) is the coefficient of kinetic friction (given as 0.250),
  • \( (m_1 + m_2) \) is the combined mass of the stuntman and villain, and
  • \( g \) is the acceleration due to gravity.

This frictional force works to slow down their slide until they come to a stop. The work done by friction is the energy dissipated, which is used to calculate the distance they slide using energy principles. By understanding frictional force, we can predict the motion and stopping distance of objects after an impact, adding another dimension to the study of physics in collisions.