We are given the equations for the positions of two cars, $A$ and $B$, as functions of time, with the following parameters:

For Car A: $x_A(t) = \alpha t + \beta t^2$ where $\alpha = 2.60 \, \text{m/s}$ and $\beta = 1.20 \, \text{m/s}^2$.

For Car B: $x_B(t) = \gamma t^2 - \delta t^3$ where $\gamma = 2.80 \, \text{m/s}^2$ and $\delta = 0.20 \, \text{m/s}^3$.

To understand their motion and how they relate to one another, we can analyze several aspects of their behavior:

1. Position of Each Car Over Time

• For Car A: $x_A(t) = 2.60t + 1.20t^2$
• For Car B: $x_B(t) = 2.80t^2 - 0.20t^3$

2. Velocities

We can differentiate the position functions with respect to time $t$ to find the velocities.

• For Car A, the velocity $v_A(t)$ is the derivative of $x_A(t)$: $v_A(t) = \frac{d}{dt} \left( 2.60t + 1.20t^2 \right) = 2.60 + 2 \times 1.20 t = 2.60 + 2.40t$ So, the velocity of Car A increases linearly with time.

• For Car B, the velocity $v_B(t)$ is the derivative of $x_B(t)$: $v_B(t) = \frac{d}{dt} \left( 2.80t^2 - 0.20t^3 \right) = 2 \times 2.80t - 3 \times 0.20t^2 = 5.60t - 0.60t^2$ Car B's velocity is more complex: it initially increases, reaches a peak, and then decreases due to the negative cubic term.

3. When Do the Cars Have the Same Position?

To find the time $t$ at which the cars are at the same position, we set $x_A(t) = x_B(t)$: $2.60t + 1.20t^2 = 2.80t^2 - 0.20t^3$ This is a cubic equation, and we can solve it to find the time(s) when the cars are at the same position.

4. When Do the Cars Have the Same Velocity?

To find when the cars have the same velocity, we set $v_A(t) = v_B(t)$: $2.60 + 2.40t = 5.60t - 0.60t^2$ This is a quadratic equation and can be solved for $t$ to find when their velocities match.

Would you like me to solve these equations for you and provide the specific time values?

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Five Related Questions:

1. How do we find the acceleration of each car at any given time?
2. What is the maximum speed Car B can achieve?
3. At what time will Car A have traveled the same distance as Car B?
4. How far from the starting point is each car after a specific time, say $t = 5 \, \text{seconds}$?
5. How can we calculate the relative velocity between Car A and Car B at a given time?

Tip:

When comparing two moving objects, analyzing their velocity and acceleration helps predict whether they will converge, diverge, or maintain a constant distance apart over time.