To solve this problem, we need to apply the principle of conservation of momentum, as this is a perfectly inelastic collision. In a perfectly inelastic collision, the two objects stick together after the collision, and their combined momentum remains conserved.

Given Data:

• Mass of Car 1 ($m_1$) = 1300 kg
• Initial Velocity of Car 1 ($v_1$) = 20 m/s (assuming this direction is positive)
• Mass of Car 2 ($m_2$) = 900 kg
• Initial Velocity of Car 2 ($v_2$) = -15 m/s (since it is moving in the opposite direction, we assign it a negative sign)

Step 1: Calculate the Total Momentum Before Collision

The total momentum before the collision is the sum of the momenta of both cars:

$\text{Total Momentum} = m_1 \cdot v_1 + m_2 \cdot v_2$

Substitute the values:

$\text{Total Momentum} = (1300 \, \text{kg}) \cdot (20 \, \text{m/s}) + (900 \, \text{kg}) \cdot (-15 \, \text{m/s})$

$\text{Total Momentum} = 26000 \, \text{kg m/s} - 13500 \, \text{kg m/s}$

$\text{Total Momentum} = 12500 \, \text{kg m/s}$

Step 2: Calculate the Combined Mass

Since the cars stick together after the collision, the combined mass ($M$) is the sum of both masses:

$M = m_1 + m_2 = 1300 \, \text{kg} + 900 \, \text{kg} = 2200 \, \text{kg}$

Step 3: Calculate the Final Velocity ($v_f$)

Since momentum is conserved, the total momentum after the collision is equal to the total momentum before the collision. The final velocity ($v_f$) of the combined mass can be calculated as follows:

$v_f = \frac{\text{Total Momentum}}{M} = \frac{12500 \, \text{kg m/s}}{2200 \, \text{kg}}$

$v_f \approx 5.68 \, \text{m/s}$

Final Answer:

The speed of the cars after the impact is approximately 5.68 m/s, and they will move in the direction of the initially moving 1300 kg car (since the final velocity is positive, matching the direction of $v_1$).

Would you like further details, or do you have any questions about the solution?

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Here are five related questions that expand on this concept:

1. What would happen to the final velocity if both cars had the same mass?
2. How would the solution change if the collision were elastic instead of inelastic?
3. If frictional forces were present, how would they impact the conservation of momentum?
4. What is the significance of momentum conservation in collisions?
5. How does the kinetic energy change in a perfectly inelastic collision?

Tip: Remember, in a perfectly inelastic collision, objects stick together post-collision, so kinetic energy is not conserved, but momentum is.