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.

Analyze the motion of two cars, A and B, using polynomial functions of time. Car A's motion is represented by a quadratic function, and Car B's motion by a cubic function. This problem covers key kinematic concepts such as velocity, acceleration, and finding when the cars meet. Suitable for high school physics and calculus students in Grades 11-12.

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