Problem 22 - College Physics

Question

\(\cdot\) On a highly polished, essentially frictionless lunch counter, a 0.500 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) Final velocity is \(-1.6\, \text{m/s}\). (b) 1.47 J dissipates as heat/sound.

Step 1: Define the conservation of momentum

For perfectly inelastic collisions (where objects stick together), the principle of conservation of momentum applies. The total momentum before the collision is equal to the total momentum after the collision. Let \( m_1 = 0.500 \, \text{kg} \) (submarine sandwich) with velocity \( v_1 = -3.00 \, \text{m/s} \) (since it's moving to the left) and \( m_2 = 0.250 \, \text{kg} \) (grilled cheese sandwich) with velocity \( v_2 = 1.20 \, \text{m/s} \) (to the right). The formula for conservation of momentum is \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \).

Step 2: Substitute the known values

Substitute \( m_1 = 0.500 \, \text{kg} \), \( v_1 = -3.00 \, \text{m/s} \), \( m_2 = 0.250 \, \text{kg} \), and \( v_2 = 1.20 \, \text{m/s} \) into the momentum equation: \[0.500 \times (-3.00) + 0.250 \times 1.20 = (0.500 + 0.250) v_f\]

Step 3: Calculate the final velocity

Simplify and solve the equation from Step 2: \[-1.5 + 0.3 = 0.750 \times v_f \-1.2 = 0.750 \times v_f \v_f = \frac{-1.2}{0.750} = -1.6 \, \text{m/s}\]Thus, the final velocity \( v_f \) of the sandwiches is \(-1.6 \, \text{m/s}\).

Step 4: Calculate initial mechanical energy

We can compute the initial mechanical energy by calculating the kinetic energy of each sandwich before the collision: - Submarine sandwich kinetic energy: \( KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \times 0.500 \times (3.00)^2 = 2.25 \, \text{J} \)- Grilled cheese sandwich kinetic energy: \( KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \times 0.250 \times (1.20)^2 = 0.18 \, \text{J} \)The total initial kinetic energy is \( KE_{i} = 2.25 + 0.18 = 2.43 \, \text{J} \).

Step 5: Calculate final mechanical energy

The final mechanical energy is the kinetic energy of the combined mass moving at \( v_f = -1.6 \, \text{m/s} \): \[KE_f = \frac{1}{2} (m_1 + m_2) v_f^2 = \frac{1}{2} \times 0.750 \times (-1.6)^2 = 0.96 \, \text{J}\]

Step 6: Determine the energy dissipated in the collision

Subtract the final kinetic energy from the initial kinetic energy to find the mechanical energy dissipated: \[\Delta KE = KE_{i} - KE_f = 2.43 - 0.96 = 1.47 \, \text{J}\]The energy dissipated is 1.47 J. This energy is transformed into other forms such as heat and sound energy during the collision.

Key Concepts

Conservation of Momentum

In physics, momentum represents the quantity of motion an object possesses. It is calculated by multiplying the object’s mass by its velocity. During a collision, the total momentum of a closed system remains constant, a principle known as the conservation of momentum. This is especially true for inelastic collisions where objects stick together after colliding. The exercise given illustrates this principle with two sandwiches colliding on a frictionless surface.

The equation for conservation of momentum during a perfectly inelastic collision is expressed as:

• Initial total momentum = Final total momentum
• This can be rewritten as: \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \)

Substituting the given values for masses and velocities, students can calculate the final velocity of the combined mass after the collision. The negative value indicates direction, showing that the final velocity is towards the left.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It is calculated using the formula \( KE = \frac{1}{2}mv^2 \). Unlike momentum, kinetic energy is not generally conserved in inelastic collisions. This is because energy is transformed into different forms during the collision process.

In the exercise, the initial kinetic energy includes contributions from both the submarine sandwich and the grilled cheese sandwich, calculated individually:

• Submarine sandwich: \( KE_1 = \frac{1}{2} (0.500 \,\text{kg}) (3.00 \,\text{m/s})^2 = 2.25 \,\text{J} \)
• Grilled cheese sandwich: \( KE_2 = \frac{1}{2} (0.250 \,\text{kg}) (1.20 \,\text{m/s})^2 = 0.18 \,\text{J} \)

The total initial kinetic energy sums to 2.43 J. After the collision, the new kinetic energy is significantly lower due to the change in movement and absorption of energy into other forms.

Mechanical Energy Dissipation

Energy dissipation refers to the process where mechanical energy is lost from a system, typically transforming into other forms such as heat, sound, or internal energy. In perfectly inelastic collisions, it is a key concept as the kinetic energy is not conserved.

The dissipated energy in the given exercise is calculated by finding the difference between the initial and final kinetic energies:

• Initial total kinetic energy: 2.43 J
• Final kinetic energy: 0.96 J
• Dissipated energy: \( \Delta KE = 2.43 \,\text{J} - 0.96 \,\text{J} = 1.47 \,\text{J} \)

This dissipated energy reflects energy transformed from motion to potentially other forms like heat due to friction rationalized at a microscopic scale and sound emitted from the collision. Understanding where the energy goes provides insight into energy conservation in practical terms.

Collision Physics

Collision physics explores the forces and interactions when two or more bodies impact each other. It analyzes how momentum and energy are transferred or transformed during these events. Inelastic collisions, like the one in the exercise, offer an insightful look into the mechanics of such interactions.

Here are key points about inelastic collisions:

• Objects stick together post-collision, moving with a common velocity.
• Momentum is conserved, but kinetic energy is not.
• Energy lost transforms into heat, sound, or deformation.

In the example, the collision demonstrates these principles. The final velocity determined through momentum conservation reflects the nature of the system post-collision. Knowledge of collisions helps in diverse fields, from designing safer vehicles to understanding cosmic events.