Question

A railroad freight car weighing \(31.8\) tons and traveling at \(5.20 \mathrm{ft} / \mathrm{s}\) overtakes one weighing \(24.2\) tons and traveling at \(2.90 \mathrm{ft} / \mathrm{s}\) in the same direction. ( \(a\) ) Find the speeds of the cars after collision if the cars couple together. \((b)\) If instead, as is very unlikely, the collision is elastic, find the speeds of the cars after collision.

Short answer

In scenario (a), both cars move together at a speed of \(4.20 \, ft/s\) after the collision. In scenario (b), assuming an elastic collision, the first car moves with a speed of \(0.648 \, ft/s\) and the second car moves with a speed of \(7.77 \, ft/s\) after the collision.

Step-by-step solution

Understand the Problem

First, we convert the weight of each car into mass (\(m\)) by dividing it by the gravitational acceleration (\(g = 32.2 ft/s^2\)). Then, we use the formula for momentum before the collision (\(P_{before} = m_1 * v_1 + m_2 * v_2\)), where \(v\) is the velocity of the cars.

Calculate Momentum before the collision

Using the formula from step 1, we calculate the initial momentum of both cars. The heavier car has mass \(m_1 = 31.8 \, tons / 32.2 \, ft/s^2 = 0.987 \, slugs\) and speed \(v_1 = 5.2 \, ft/s\). The lighter car has mass \(m_2 = 24.2 \, tons / 32.2 \, ft/s^2 = 0.751 \, slugs\) and speed \(v_2 = 2.9 \, ft/s\). Thus, \(P_{before} = m_1 * v_1 + m_2 * v_2 = 0.987 \, slugs * 5.2 \, ft/s + 0.751 \, slugs * 2.9 \, ft/s = 7.215 \, slug*ft/s\)

Compute the Velocity after the collision (Scenario a)

Next, we compute the velocity of the two cars after they couple together using the law of conservation of momentum, thus \(v_{after}= P_{before} / (m_1 + m_2) = 7.215 \, slug*ft/s / (0.987 \, slugs + 0.751 \, slugs) = 4.20 \, ft/s\)

Compute the Velocities after the collision (Scenario b)

For an elastic collision, both kinetic energy and momentum are conserved. The kinetic energy before the collision is \(KE_{before} = 1/2 * m_1 * v_1^2 + 1/2 * m_2 * v_2^2\). We solve the two equations \(P_{before} = m_1 * v_1' + m_2 * v_2'\) and \(KE_{before} = 1/2 * m_1 * v_1'^2 + 1/2 * m_2 * v_2'^2\) simultaneously, with \(v_1'\) and \(v_2'\) being the velocities after the collision. Solving this gives us \(v_1' = 0.648 \, ft/s\) and \(v_2' = 7.77 \, ft/s\)

Key concepts

Conservation of Momentum

When we study the motion and interaction of objects, one of the most fundamental principles is the conservation of momentum. This principle states that within an isolated system, the total momentum remains constant over time, unless acted upon by external forces. Momentum is a vector quantity, which means it has both magnitude and direction, defined by the equation \(P = m \cdot v\), where \(m\) is the mass of the object and \(v\) is its velocity.

Imagine two railway freight cars, one heavier and one lighter, careening down tracks and eventually colliding. If they couple and move together, we can predict their combined velocity by applying conservation of momentum. We take the combined mass of the two cars and set the total momentum before the collision equal to the total momentum after the collision to find the velocity. This approach simplifies complex scenarios into solvable problems, helping students understand that in physics, momentum links the conditions before and after an event in a perfectly predictable way.

Elastic Collision

An elastic collision is a specific type of collision where not only is the total momentum conserved, but the total kinetic energy of the objects involved is also conserved. Kinetic energy is the energy an object has due to its motion and can be calculated with the formula \(KE = \frac{1}{2} m v^2\).

In an elastic collision, objects collide and rebound without losing energy to deformation or heat. Imagine two freight cars hitting and rebounding without coupling. To find their speeds after collision, we'd employ both conservation of momentum and kinetic energy.

Dissecting the Unlikely Scenario

Although elastic collisions between large objects like freight cars are unlikely in the real world due to energy losses, they are a useful theoretical concept in physics. They allow us to delve into the behavior of particles at a microscopic level, where such collisions are more common, and also to understand maximum theoretical efficiency in energy transfer between objects.

Kinetic Energy Conservation

Kinetic energy conservation in an elastic collision is a remarkable phenomenon that charts the behavior of objects in a perfectly efficient system. When two objects collide and no kinetic energy is lost to sound, heat, or any other form of energy, we say that the kinetic energy is conserved. It's integral to solving collision problems, as it provides a second equation to work with alongside the conservation of momentum.

To explore this concept in our train example, if the collision were elastic, after computing the initial kinetic energies of both cars, we would set the sum of their kinetic energies before the collision equal to the sum after the collision. This constraint, combined with momentum conservation, allows us to solve for the unknown velocities post-collision.

Relevance in Real-World Applications

Understanding kinetic energy conservation is crucial, not only for theoretical physics problems but also for designing safer vehicles, improving athletic gear, and creating more efficient machinery, where managing impact forces and energy dissipation is key.