Problem 21 - College Physics

Question

2On a highly polished, essentially frictionless lunch counter, a \(0.500 \mathrm{~kg}\) submarine sandwich moving \(3.00 \mathrm{~m} / \mathrm{s}\) to the left collides with a \(0.250 \mathrm{~kg}\) grilled cheese sandwich moving \(1.20 \mathrm{~m} / \mathrm{s}\) to the right. (a) If the two sandwiches stick together, what is their final velocity? (b) How much mechanical energy dissipates in the collision? Where did this energy go?

Step-by-Step Solution

Answer

(a) The final velocity is -1.60 m/s to the left. (b) 1.47 J of energy dissipates, transformed into heat and sound.

Step 1: Understand the Conservation of Momentum

The total momentum before the collision must equal the total momentum after the collision due to the law of conservation of momentum. This principle is expressed by the formula:\[ m_1 \cdot v_{1i} + m_2 \cdot v_{2i} = (m_1 + m_2) \cdot v_f \], where\( m_1 \) and \( v_{1i} \) are the mass and initial velocity of the first sandwich (submarine), \( m_2 \) and \( v_{2i} \) are the mass and initial velocity of the second sandwich (grilled cheese), and \( v_f \) is the final velocity of both sandwiches when they stick together.

Step 2: Substitute Known Values

Substitute the given values into the conservation of momentum equation:Mass of submarine sandwich, \( m_1 = 0.500 \, \text{kg} \), velocity \( v_{1i} = -3.00 \, \text{m/s} \) (left is negative) Mass of grilled cheese sandwich, \( m_2 = 0.250 \, \text{kg} \), velocity \( v_{2i} = 1.20 \, \text{m/s} \)\[ 0.500 \times (-3.00) + 0.250 \times 1.20 = (0.500 + 0.250) \cdot v_f \]

Step 3: Solve for Final Velocity (v_f)

Calculate both sides of the equation:\[ -1.50 + 0.30 = 0.750 \cdot v_f \] \[ -1.20 = 0.750 \cdot v_f \]Solving for \( v_f \):\[ v_f = \frac{-1.20}{0.750} = -1.60 \, \text{m/s} \] The negative sign indicates that the final velocity is to the left.

Step 4: Calculate Initial Mechanical Energy

Calculate the initial kinetic energy of each sandwich.Initial kinetic energy of the submarine sandwich:\[ KE_{1i} = \frac{1}{2} m_1 v_{1i}^2 = \frac{1}{2} (0.500)(3.00)^2 = 2.25 \, \text{J} \]Initial kinetic energy of the grilled cheese sandwich:\[ KE_{2i} = \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} (0.250)(1.20)^2 = 0.18 \, \text{J} \]Total initial kinetic energy:\[ KE_{initial} = 2.25 + 0.18 = 2.43 \, \text{J} \]

Step 5: Calculate Final Mechanical Energy

Calculate the kinetic energy of both sandwiches stuck together moving at \( v_f = -1.60 \, \text{m/s} \):\[ KE_{final} = \frac{1}{2} (m_1 + m_2) (v_f)^2 = \frac{1}{2} (0.750) (-1.60)^2 = 0.96 \, \text{J} \]

Step 6: Calculate Mechanical Energy Loss

The difference between initial and final kinetic energy gives the energy dissipated:\[ E_{dissipated} = KE_{initial} - KE_{final} = 2.43 - 0.96 = 1.47 \, \text{J} \]This energy likely transformed into heat and sound during the collision.

Key Concepts

Elastic Collisions

In the fascinating world of physics, a collision refers to a significant event where two or more bodies exert force upon one another in a relatively short period of time. Elastic collisions are a type of collision where not only is momentum conserved, but kinetic energy is also preserved before and after the event.
These contrasts differ from inelastic collisions, where the objects might stick together, leading to a loss of kinetic energy, often transformed into other energy forms, like heat or sound.
In the given problem involving the sandwiches, we consider them sticking together, which means they undergo an inelastic collision. If it were an elastic collision, after the interaction, the masses would separate with no energy lost to the surroundings.
For students, it's essential to understand the distinction between these two types and the conditions under which the mechanical properties, like velocity, are calculated in each case. Elastic collisions are idealized scenarios, often studied in gas particle interactions because real-life collisions, akin to our sandwich example, tend to involve some form of energy dissipation.

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. The three primary factors affecting it are the mass of the object, its velocity, and the direction of its motion (considering vectors for cases involving direction).
In mathematical terms, kinetic energy is calculated using the formula: \[ KE = \frac{1}{2} m v^2 \]
where \( m \) stands for mass and \( v \) is velocity. In the context of the exercise, the initial kinetic energy of each sandwich was calculated individually before they collided.
After the collision, where both sandwiches stick together, we calculate the combined kinetic energy at the resultant final velocity. During this calculation, you often find that fewer joules are exhibited in the final kinetic energy compared to the initial phase before collision.
This discrepancy is crucial as it leads us into understanding mechanical energy loss—indicating the non-conserved nature of kinetic energy during inelastic collisions like the one described here, where energy conversion occurs.

Mechanical Energy Loss

When discussing mechanical energy, it comprises both kinetic and potential energy. In our sandwich collision scenario, mechanical energy loss primarily occurs as kinetic energy transforms into other forms.
Calculations revealed a decrease in mechanical energy from initial to final states, demonstrating a loss of 1.47 joules. This energy doesn't simply disappear; it converts into other forms of energy, predominately heat and possibly sound, when the collisions aren't perfectly elastic.
This transformation highlights the practical and often unavoidable nature of real-world collisions.
Understanding this can lead students to realize that even when the total energy is conserved (in accordance with the Law of Conservation of Energy), not all of it remains in a form that's useful for doing mechanical work.
Thus, in the context of the everyday world where friction and other forces exist, the balance of energies shifts, with some mechanical energy dissipated into other forms, explaining the seemingly missing energy post-collision.