Problem 29 - Physics for Scientists and Engineers with Modern Physics

Question

(I) Two railroad cars, each of mass \(56,000 \mathrm{~kg}\), are traveling \(95 \mathrm{~km} / \mathrm{h}\) toward each other. They collide head-on and come to rest. How much thermal energy is produced in this collision?

Step-by-Step Solution

Answer

The thermal energy produced is the initial kinetic energy of both cars.

Step 1: Convert Speed to Meters Per Second

The railroad cars are traveling at 95 km/h, which needs to be converted to meters per second (m/s). To do this, use the conversion factor: 1 km/h = 0.27778 m/s. So, \[ 95 \text{ km/h} \times 0.27778 \text{ m/s per km/h} = 26.39 \text{ m/s} \]

Step 2: Calculate Initial Kinetic Energy per Car

The initial kinetic energy for each car can be calculated using the kinetic energy formula: \[ KE = \frac{1}{2} mv^2 \]where \( m = 56000 \, \text{kg} \) and \( v = 26.39 \text{ m/s} \). \[ KE = \frac{1}{2} \times 56000 \times (26.39)^2 \]Calculate the value to find the initial kinetic energy per car.

Step 3: Find Total Initial Kinetic Energy

Since there are two cars, the total initial kinetic energy is twice the kinetic energy calculated for one car. Multiply the result from the previous step by 2.

Step 4: Determine Final Kinetic Energy

After the collision, both cars come to rest, resulting in zero kinetic energy. Thus, the final kinetic energy is \( 0 \text{ J} \).

Step 5: Calculate Thermal Energy Produced

The thermal energy produced in the collision is equal to the total initial kinetic energy because all the kinetic energy is converted into thermal energy as the cars come to rest. Thermal energy produced = Total Initial Kinetic Energy.

Key Concepts

Kinetic Energy

Kinetic energy is a fascinating concept that describes the energy an object has due to its motion. It can be likened to the power you feel when you ride a bike fast downhill. In this specific scenario, each railroad car is traveling at a speed of 26.39 meters per second—which is the after conversion of 95 km/h. In physics, the formula for kinetic energy is expressed as:
\[ KE = \frac{1}{2} mv^2 \]where:

• \( m \) is the mass of the object (in kilograms)
• \( v \) is the velocity of the object (in meters per second)
• \( KE \) stands for kinetic energy (in joules)

In the case of the railroad cars, with a mass of 56,000 kg each, the calculation of kinetic energy would show us how much energy each car has due to its motion before colliding. This helps quantify the energy transformation during the collision.

Thermal Energy

Thermal energy is the energy that results from the motion of particles within a substance—often perceived as heat. When the two railroad cars collide and come to rest, their initial kinetic energy doesn't vanish. Instead, it's transformed into thermal energy and possibly some other forms like sound or deformation.
The principle of conservation of energy dictates that the total energy of a closed system remains constant. Therefore, the initial kinetic energy of the railroad cars, prior to the collision, is entirely converted into thermal energy, given that they come to rest and all kinetic energy is zero at the end:

• Initial Kinetic Energy: Energy due to motion.
• Converted Thermal Energy: The resulting heat energy.

By understanding this conversion process, it reveals how energy transformations play an integral role in everyday events and phenomena!

Collision Analysis

Collision analysis is crucial in physics as it helps understand what occurs during and after the impact of two or more objects. In this head-on collision between the railroad cars, several critical insights can be extracted:

• The cars start with a significant amount of kinetic energy, moving towards each other.
• Upon collision, they engage in an exchange and transformation of energy as they come to a complete halt.
• The kinetic energy is not really 'lost'. Instead, it is converted entirely into other forms, predominantly as thermal energy.
• This transformation adheres to the conservation of energy principle, reaffirming energy’s persistent nature but changing form.

Collision analysis, thus, doesn't just help in calculating the aftermath of physical interactions but also in predicting how energy exchanges and transformations occur in various systems.