91Ó°ÊÓ

Kinetic friction is a force that opposes the motion of two surfaces sliding past each other. In this exercise, the kinetic friction comes into play when calculating the force that will slow down the coupled railroad cars. It is crucial in determining how long it will take for the cars to come to a complete stop. Kinetic friction is calculated using the formula:

• \( f_k = \mu_k \cdot N \)

where \( f_k \) is the frictional force, \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force exerted by the objects in contact.

In this scenario, the normal force is the combined weight of both railroad cars, which totals 150,000 pounds. By multiplying this weight by the kinetic friction coefficient of 0.30, we find the frictional force to be 45,000 pounds. Understanding kinetic friction helps us know how much force is resisting the motion, and how it contributes to eventually stopping the cars.

During the coupling of railroad cars, conservation of momentum is the primary principle used. When two cars are brought together, the moving car transfers some of its momentum to the stationary car, causing both to move as one unit. This concept relies on the law of conservation of momentum, which can be stated as the total momentum before the coupling must equal the total momentum after the coupling. This is because no external horizontal forces act on the system during coupling.

In practice, this is expressed by the equation:

• \[ p_{initial\,system} = p_{final\,system} \]
• \[ m_1v_1 = (m_1 + m_2)v_f \]

Where \( m_1 \) and \( m_2 \) are the masses of the first and second cars, respectively, and \( v_f \) is their final velocity after coupling. Using this principle, we can determine the final velocity of the two cars, which is found to be approximately 1.223 ft/s in the exercise. Understanding this helps predict the combined behavior of objects after interactions without external interference.

Deceleration occurs when the velocity of an object decreases over time, which is a form of acceleration but in the opposite direction of the movement. In this scenario, deceleration happens because of the kinetic friction force acting on the coupled railroad cars. To calculate deceleration, we use the formula:

• \[ f_k = m \cdot a \]

where \( f_k \) is the kinetic friction force, \( m \) is the total mass of the system (both railroad cars), and \( a \) is the deceleration.

From the exercise, the calculated deceleration is approximately \( 9.66 \text{ ft/s}^2 \). Using this deceleration value, we can find out how long it takes for the railroad cars to come to a stop. By applying the equation of motion:

• \[ v = v_0 - at \]

where \( v \) is the final velocity (0 ft/s when cars come to a stop), \( v_0 \) is the initial velocity, and \( t \) is the time taken. Solving the equation gives us a time of about 0.1267 seconds for the stop. Deceleration is essential in understanding how quickly systems can come to rest under opposing forces.