Two cars, A and B, travel in a straight line. The distance of car A from the starting point is given as a function of time by xA(t) = αt + βt^2, with α = 2.60 m/s and β = 1.20 m/s^2. The distance of car B from the starting point is xB(t) = γt^2 - δt^3, with γ = 2.80 m/s^2 and δ = 0.20 m/s^3.

Given the functions describing the distance of cars A and B from the starting point as functions of time \( t \):

- For car A:  
  \[
  x_A(t) = \alpha t + \beta t^2
  \]
  where \( \alpha = 2.60 \, \text{m/s} \) and \( \beta = 1.20 \, \text{m/s}^2 \).  
  Substituting the values:
  \[
  x_A(t) = 2.60t + 1.20t^2
  \]

- For car B:  
  \[
  x_B(t) = \gamma t^2 - \delta t^3
  \]
  where \( \gamma = 2.80 \, \text{m/s}^2 \) and \( \delta = 0.20 \, \text{m/s}^3 \).  
  Substituting the values:
  \[
  x_B(t) = 2.80t^2 - 0.20t^3
  \]

Here are some possible questions we can explore:

1. **At what time do the two cars meet?** This involves solving the equation \( x_A(t) = x_B(t) \).

2. **What are the velocities of both cars as a function of time?** This requires finding the first derivative of \( x_A(t) \) and \( x_B(t) \) with respect to time.

3. **What are the accelerations of both cars as a function of time?** This involves finding the second derivative of \( x_A(t) \) and \( x_B(t) \).

4. **What is the distance between the two cars at a specific time \( t \)?** This can be found using \( |x_A(t) - x_B(t)| \).

5. **What is the relative velocity of car A with respect to car B at a specific time \( t \)?** This is found by taking the difference in their velocities.

Would you like to explore any of these questions or have more details on the problem? 

**Tip:** When comparing two functions, setting them equal and solving for the variable can reveal critical points, such as when two moving objects meet.

This problem explores the motion of two cars with distance functions involving quadratic and cubic equations. Learn how to find when the cars meet, calculate their velocities, accelerations, and relative positions. Suitable for high school and college physics students.

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