A minibus taxi with an unknown number of passengers travels at $$ 25 \mathrm{~m} \cdot \mathrm{~s}^{-1} $$ when it collides with a car with a mass of 1200 kg (passengers included), travelling at $$ 15 \mathrm{~m} \cdot \mathrm{~s}^{-1} $$ in the opposite direction, as shown. During the collision the vehicles stick together and travel at $$ 14 \mathrm{~m} \cdot \mathrm{~s}^{-1} $$ immediately after the collision in the direction of the original motion of the taxi. 6.1 Ignore friction. Use momentum principles to determine whether the minibus taxi was overloaded, that is, above the legal combined mass of 3500 kg . 6.2 Is the collision betiveen the vehicles elastic or inelastic? Support your answer with an appropriate calculation.

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### Problem 6.1: Determining if the Minibus Taxi Was Overloaded

To determine whether the minibus taxi was overloaded, we'll use the principle of **conservation of momentum**.

**Given:**
- **Mass of the car (including passengers):** $$ m_c = 1200 \, \text{kg} $$
- **Velocity of the minibus taxi before collision:** $$ v_m = 25 \, \text{m/s} $$ (positive direction)
- **Velocity of the car before collision:** $$ v_c = -15 \, \text{m/s} $$ (negative direction, opposite to the minibus)
- **Combined velocity after collision:** $$ v' = 14 \, \text{m/s} $$ (in the direction of the minibus)

**Let:** 
- $$ m_m $$ be the mass of the minibus taxi (including passengers).

**Applying Conservation of Momentum:**
$$
\text{Total Momentum Before Collision} = \text{Total Momentum After Collision}
$$
$$
m_m \cdot v_m + m_c \cdot v_c = (m_m + m_c) \cdot v'
$$
$$
m_m \cdot 25 + 1200 \cdot (-15) = (m_m + 1200) \cdot 14
$$
$$
25m_m - 18000 = 14m_m + 16800
$$
$$
25m_m - 14m_m = 16800 + 18000
$$
$$
11m_m = 34800
$$
$$
m_m = \frac{34800}{11} \approx 3163.64 \, \text{kg}
$$

**Conclusion:**
- **Legal Combined Mass Limit:** $$ 3500 \, \text{kg} $$
- **Calculated Combined Mass:** $$ m_m + m_c = 3164 \, \text{kg} + 1200 \, \text{kg} = 4364 \, \text{kg} $$

The calculated combined mass of **4364 kg** exceeds the legal combined mass limit of **3500 kg**. However, it's important to clarify whether the legal limit refers to the minibus alone or the combined mass of both vehicles. If it's solely for the minibus, then the minibus mass (**3164 kg**) is **below** the limit. Assuming the legal limit pertains to the minibus alone:

**The minibus taxi was **not** overloaded**, as its mass of approximately **3164 kg** is below the legal limit of **3500 kg**.

---

### Problem 6.2: Determining if the Collision Was Elastic or Inelastic

A collision is **elastic** if both momentum and **kinetic energy** are conserved. If only momentum is conserved but kinetic energy is not, the collision is **inelastic**.

**Calculating Kinetic Energy Before Collision:**
$$
KE_{\text{before}} = \frac{1}{2} m_m v_m^2 + \frac{1}{2} m_c v_c^2
$$
$$
KE_{\text{before}} = \frac{1}{2} \times 3164 \times 25^2 + \frac{1}{2} \times 1200 \times 15^2
$$
$$
KE_{\text{before}} = 0.5 \times 3164 \times 625 + 0.5 \times 1200 \times 225
$$
$$
KE_{\text{before}} = 988750 \, \text{J} + 135000 \, \text{J} = 1,123,750 \, \text{J}
$$

**Calculating Kinetic Energy After Collision:**
$$
KE_{\text{after}} = \frac{1}{2} (m_m + m_c) v'^2
$$
$$
KE_{\text{after}} = \frac{1}{2} \times 4364 \times 14^2
$$
$$
KE_{\text{after}} = 0.5 \times 4364 \times 196 = 427,672 \, \text{J}
$$

**Comparison:**
$$
KE_{\text{before}} = 1,123,750 \, \text{J} \quad \text{vs.} \quad KE_{\text{after}} = 427,672 \, \text{J}
$$

**Conclusion:**
The kinetic energy **decreased** during the collision, indicating that **not all kinetic energy was conserved**.

**Therefore, the collision was **inelastic**.
The minibus taxi was overloaded because its combined mass with the car exceeded the legal limit. The collision was inelastic since kinetic energy was not conserved during the collision.

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