Problem 21 - College Physics

Question

\(\cdot\) Three identical boxcars are coupled together and are moving at a constant speed of 20.0 \(\mathrm{m} / \mathrm{s}\) on a level track. They collide with another identical boxcar that is initially at rest and couple to it, so that the four cars roll on as a unit. Friction is small enough to be neglected. (a) What is the speed of the four cars? (b) What percentage of the kinetic energy of the boxcars is dissipated in the collision? What happened to this energy?

Step-by-Step Solution

Answer

(a) The speed is 15.0 m/s. (b) 25% of the kinetic energy is dissipated as heat and sound.

Step 1: Understanding the Initial Momentum

The momentum before the collision is the momentum of the three moving boxcars, as the fourth boxcar is initially at rest. The total mass of these three boxcars is \(3m\) (where \(m\) is the mass of one boxcar). Their velocity is 20.0 \(\mathrm{m/s}\).The initial momentum \( p_i \) is given by \[ p_i = 3m \times 20.0 \, \text{m/s} = 60m \, \text{m/s}. \]

Step 2: Calculating the Final Momentum

After the collision, the four boxcars move together. The total mass now is \(4m\). We denote \(v_f\) as the final speed of the four cars and use conservation of momentum.The equation for final momentum \( p_f \) is:\[ p_f = 4m \times v_f. \]

Step 3: Applying Conservation of Momentum

According to the law of conservation of momentum, the total momentum before the collision equals the total momentum after the collision.Thus, \[ 60m = 4m \times v_f. \]Cancelling \(m\) from both sides gives:\[ v_f = \frac{60}{4} = 15.0 \, \text{m/s}. \]

Step 4: Calculating Initial Kinetic Energy

The initial kinetic energy \(KE_i\) is only from the three moving cars:\[ KE_i = \frac{1}{2} \times 3m \times (20.0)^2 = 600m \, \text{J}. \]

Step 5: Calculating Final Kinetic Energy

The kinetic energy after the collision, for the four cars moving together, is:\[ KE_f = \frac{1}{2} \times 4m \times (15.0)^2 = 450m \, \text{J}. \]

Step 6: Determining Kinetic Energy Dissipation

The energy dissipated is the difference between the initial and final kinetic energies:\[ \Delta KE = KE_i - KE_f = 600m - 450m = 150m \, \text{J}. \]

Step 7: Calculating Energy Dissipation Percentage

The percentage of the initial kinetic energy that is dissipated is found by:\[ \frac{\Delta KE}{KE_i} \times 100\% = \frac{150m}{600m} \times 100\% = 25\%. \]

Step 8: Explaining Energy Dissipation

The kinetic energy lost in the collision is transformed into other forms of energy, primarily heat and sound, as well as some deformation work on the boxcars.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. For an object with mass, this energy can be defined using the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the object's mass and \( v \) is its velocity.

In the context of our exercise, the three moving boxcars initially possess kinetic energy due to their speed of 20.0 m/s. This energy allows them to move along the track, and when they collide with the fourth boxcar, the total kinetic energy is redistributed.

• Initial kinetic energy is calculated for the three moving boxcars.
• Final kinetic energy is recalculated after collision and coupling.

The difference in these values indicates the energy dissipated during the collision.

Inelastic Collisions

During an inelastic collision, the objects involved do not retain their total kinetic energy, although the system's total momentum is conserved. This is a key characteristic of an inelastic collision.

In our case, the boxcars collide and stick together, forming a single, combined mass. This behavior showcases a perfectly inelastic collision, where the maximum possible kinetic energy is transformed into other forms of energy. The fact that the boxcars move as one after the collision points to this energy conversion mechanism.

• Kinetic energy is not conserved in inelastic collisions.
• The objects may change shape or produce heat and sound, leading to energy dissipation.

Momentum

Momentum is a measure of the motion of a body and is the product of its mass and velocity \( p = mv \). It is a vector quantity, possessing both direction and magnitude.

The conservation of momentum principle states that in a closed system, without external forces, the total momentum before any event will equal the total momentum after it.

• The initial momentum in this exercise is derived from the three-moving boxcars.
• Post-collision, the momentum is distributed across the four boxcars, maintaining the system's total momentum.

The conservation law allows us to calculate the final speed of the coupled boxcars in the given scenario.

Energy Dissipation

Energy dissipation refers to the process where useful energy (like kinetic energy) is transformed into a less useful form, such as heat or sound. In physical systems, this energy transformation is usually unavoidable.

In our exercise, the 25% dissipation of initial kinetic energy indicates a transformation into non-mechanical forms during the boxcar collision.

• Dissipated energy often contributes to warming up the involved bodies or the surrounding environment.
• Some of the energy might also be used in permanent deformation of the colliding materials.

Understanding energy dissipation is crucial in assessing efficiency and performance in mechanical systems.