Question

Two railroad cars, each of mass \(7000 . \mathrm{kg}\) and traveling at \(90.0 \mathrm{~km} / \mathrm{h},\) collide head on and come to rest. How much mechanical energy is lost in this collision?

Short answer

Answer: The mechanical energy lost during the collision is 1750000 J (joules).

Step-by-step solution

Convert velocities to m/s

First, we need to convert the given velocities from km/h to m/s for both railroad cars. To do this, we can use the conversion factor 1 km/h = (1000 m/km) / (3600 s/h). Multiply the given velocity (90.0 km/h) by this conversion factor: Velocity_car1 = Velocity_car2 = 90.0 km/h × (1000 m/km) / (3600 s/h) = 25.0 m/s

Calculate the initial and final momentum of the system

Before the collision, each car has an equal and opposite momentum (since they have equal masses and velocities). The initial momentum of the system is the sum of the momentums of both cars, while the final momentum of the system is zero (since they come to rest after the collision). Initial_momentum = Mass_car1 × Velocity_car1 - Mass_car2 × (-Velocity_car2) = 7000 kg × 25.0 m/s + 7000 kg × 25.0 m/s = 350000 kg m/s Final_momentum = 0 kg m/s (since both cars come to rest)

Calculate the initial and final kinetic energy of the system

Calculate the initial kinetic energy (KE) of the system as the sum of the kinetic energies of both cars. Then calculate the final kinetic energy of the system, which is zero (since the cars come to rest). Initial_KE = 0.5 × Mass_car1 × Velocity_car1^2 + 0.5 × Mass_car2 × (-Velocity_car2)^2 = 0.5 * 7000 kg × (25.0 m/s)^2 + 0.5 × 7000 kg × (25.0 m/s)^2 = 1750000 J Final_KE = 0 J (since both cars come to rest)

Calculate the mechanical energy lost during the collision

The mechanical energy lost in the collision is equal to the difference between the initial kinetic energy and the final kinetic energy. Mechanical_energy_lost = Initial_KE - Final_KE = 1750000 J - 0 J = 1750000 J Therefore, the mechanical energy lost during the collision is 1750000 J (joules).

Key concepts

Kinetic Energy

Kinetic energy is a measure of the energy of motion. It tells us how much energy an object has due to its movement. For an object with mass \(m\) and velocity \(v\), the kinetic energy \(KE\) is given by the formula: \[ KE = \frac{1}{2}mv^2 \]In day-to-day life, this concept shows up in things like moving cars or falling objects: the faster they move, or the heavier they are, the more kinetic energy they hold.
In a collision, like with the railroad cars in our exercise, calculating the initial and final kinetic energy helps us understand how much energy was involved before and after a collision. During the head-on collision, both cars, each traveling at 25 m/s, contribute to the system's total energy. After coming to rest, their final kinetic energy drops to zero because there is no motion.
This significant change in kinetic energy indicates where and how energy transformation or loss occurs in dynamics.

Momentum Conservation

Momentum conservation is a core principle in physics stating that in an isolated system, the total momentum before a collision equals the total momentum after the collision. Momentum, represented by \(p\), is calculated using: \[ p = mv \]
In events like a collision between two objects, individual momenta might change, but the total system momentum remains constant unless external forces come into play.

• Before collision: The total momentum is the sum of the momentum of both railroad cars. They have equal masses and equal but opposite velocities, resulting in a significant combined momentum of 350,000 kg m/s.
• After collision: The system's momentum is zero as both cars come to rest, conserving momentum through the conversion of kinetic energy to other forms.

This preserved total momentum is crucial in explaining collisions, ensuring all changes in the system's motion are accounted for, even when velocities change drastically.

Energy Conversion

Energy conversion refers to the process of changing energy from one form to another. During inelastic collisions, like the one between the two railroad cars, mechanical energy gets converted mainly into thermal energy and sound.
Initially, both cars possess a considerable amount of kinetic energy, but when they collide and come to a halt, their kinetic energy must convert into other forms due to the conservation of energy.

• Thermal energy: As the cars grind against each other, friction causes an increase in temperature, resulting in thermal energy.
• Sound energy: The noise generated by the collision is a release of energy in the form of sound waves.

Therefore, the mechanical energy loss calculated in this exercise is a measure of how significant an energy conversion event the collision was, highlighting the transformation from the system's controlled kinetic state to a dissipative one.

Inelastic Collision

An inelastic collision is where colliding objects stick together post-collision, often resulting in the loss of kinetic energy. Unlike elastic collisions, where objects bounce off with no kinetic energy loss, inelastic collisions see a significant energy transformation.
In the case of the two railroad cars, they collide and come to a complete stop, which is a classic example of a perfectly inelastic collision. Here, the main features include:

• Energy Loss: There's a noticeable loss of kinetic energy which gets converted into other forms like heat and sound as seen from our calculation of 1,750,000 J lost.
• Objects Coalesce: The cars stick together or start moving together as a result of the mechanical forces during collision.

This energy conversion and sticking together phenomena illustrate how inelastic collisions differ from those we describe as elastic, where energy and momentum carry on relatively undisturbed.