A railroad freight car rolls on a track at 2.80 m/s toward two identical coupled freight cars, which are rolling in the same direction as the first, but at a speed of 1.20 m/s. The first reaches the second two and all couple together. The mass of each is 2.20 ✕ 104 kg.
(a) What is the speed (in m/s) of the three coupled cars after the first couples with the other two? (Round your answer to at least two decimal places.)
  
(b) Find the (absolute value of the) amount of kinetic energy (in J) converted to other forms during the collision.

Let's solve the problem step by step.

### (a) Speed of the Three Coupled Cars

We'll use the principle of conservation of momentum. The momentum before the collision should be equal to the momentum after the collision.

The mass of each freight car is \( m = 2.20 \times 10^4 \) kg.

**Initial momentum:**

- The first freight car is moving at \( v_1 = 2.80 \) m/s.
- The two coupled freight cars are moving at \( v_2 = 1.20 \) m/s.

The total mass of the two coupled cars is \( 2m = 2 \times 2.20 \times 10^4 \) kg = \( 4.40 \times 10^4 \) kg.

Initial momentum \( p_i \) is the sum of the momenta of the individual masses:
\[ p_i = m v_1 + 2m v_2 \]
\[ p_i = (2.20 \times 10^4 \, \text{kg} \times 2.80 \, \text{m/s}) + (4.40 \times 10^4 \, \text{kg} \times 1.20 \, \text{m/s}) \]
\[ p_i = 6.16 \times 10^4 \, \text{kg} \cdot \text{m/s} + 5.28 \times 10^4 \, \text{kg} \cdot \text{m/s} \]
\[ p_i = 11.44 \times 10^4 \, \text{kg} \cdot \text{m/s} \]

**Final momentum:**

Let \( v_f \) be the final speed of the three coupled cars. The total mass after collision is \( 3m \):
\[ 3m = 3 \times 2.20 \times 10^4 \, \text{kg} = 6.60 \times 10^4 \, \text{kg} \]

Using the conservation of momentum:
\[ p_i = p_f \]
\[ 11.44 \times 10^4 \, \text{kg} \cdot \text{m/s} = 6.60 \times 10^4 \, \text{kg} \times v_f \]
\[ v_f = \frac{11.44 \times 10^4 \, \text{kg} \cdot \text{m/s}}{6.60 \times 10^4 \, \text{kg}} \]
\[ v_f = \frac{11.44}{6.60} \, \text{m/s} \]
\[ v_f \approx 1.73 \, \text{m/s} \]

So, the speed of the three coupled cars after the collision is approximately **1.73 m/s**.

### (b) Kinetic Energy Loss

We need to calculate the kinetic energy before and after the collision to find the loss.

**Initial Kinetic Energy:**
\[ KE_i = \frac{1}{2} m v_1^2 + \frac{1}{2} (2m) v_2^2 \]
\[ KE_i = \frac{1}{2} (2.20 \times 10^4 \, \text{kg}) (2.80 \, \text{m/s})^2 + \frac{1}{2} (4.40 \times 10^4 \, \text{kg}) (1.20 \, \text{m/s})^2 \]
\[ KE_i = \frac{1}{2} (2.20 \times 10^4 \, \text{kg}) \times 7.84 \, \text{m}^2/\text{s}^2 + \frac{1}{2} (4.40 \times 10^4 \, \text{kg}) \times 1.44 \, \text{m}^2/\text{s}^2 \]
\[ KE_i = (1.10 \times 10^4 \, \text{kg}) \times 7.84 \, \text{m}^2/\text{s}^2 + (2.20 \times 10^4 \, \text{kg}) \times 1.44 \, \text{m}^2/\text{s}^2 \]
\[ KE_i = 8.624 \times 10^4 \, \text{J} + 3.168 \times 10^4 \, \text{J} \]
\[ KE_i = 11.792 \times 10^4 \, \text{J} \]

**Final Kinetic Energy:**
\[ KE_f = \frac{1}{2} (3m) v_f^2 \]
\[ KE_f = \frac{1}{2} (6.60 \times 10^4 \, \text{kg}) (1.73 \, \text{m/s})^2 \]
\[ KE_f = \frac{1}{2} (6.60 \times 10^4 \, \text{kg}) \times 2.9929 \, \text{m}^2/\text{s}^2 \]
\[ KE_f = (3.30 \times 10^4 \, \text{kg}) \times 2.9929 \, \text{m}^2/\text{s}^2 \]
\[ KE_f = 9.87657 \times 10^4 \, \text{J} \]

**Kinetic Energy Loss:**
\[ \Delta KE = KE_i - KE_f \]
\[ \Delta KE = 11.792 \times 10^4 \, \text{J} - 9.87657 \times 10^4 \, \text{J} \]
\[ \Delta KE = 1.91543 \times 10^4 \, \text{J} \]

So, the absolute value of the amount of kinetic energy converted to other forms during the collision is approximately **19,154.3 J**.

Do you want details or have any questions?

Here are some more questions you might find interesting:
1. How to calculate the impact force of two colliding objects?
2. How does conservation of energy apply in inelastic collisions?
3. What are the different types of collisions in physics?
4. How to find the final velocities in an elastic collision?
5. What is the coefficient of restitution and how is it used?

**Tip:** Always ensure to check if a collision problem is elastic or inelastic as it significantly changes how you approach the problem.

Explore how to calculate the final speed and the amount of kinetic energy converted during the collision of three coupled railroad freight cars. Understand the principles of conservation of momentum and kinetic energy in inelastic collisions. Suitable for high school physics students.

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