Question

1. lissette is driving to meet her family in a neighboring city. she accelerates at a constant rate and then maintains this speed for a period of time. an increase in traffic causes her to decelerate at a constant rate. lisette then reaches a constant speed and maintains this rate until she exits the highway. sketch a graph to represent the situation.

Explanation:

Step1: Analyze acceleration phase

When accelerating at a constant rate, the speed - time graph is a straight line with a positive slope. Let the \(y\) - axis be speed (\(v\)) and the \(x\) - axis be time (\(t\)). If the initial speed \(v_0 = 0\) (assuming she starts from rest), and the acceleration \(a>0\), the equation of the line is \(v=at\) (for the acceleration phase).

Step2: Analyze constant - speed phase

When maintaining a constant speed, the slope of the speed - time graph is \(0\). So, it is a horizontal line. If the speed after acceleration is \(v = v_1\), the equation of this part of the graph is \(v = v_1\) (for the time interval of constant - speed).

Step3: Analyze deceleration phase

When decelerating at a constant rate, the slope of the speed - time graph is negative. Let the deceleration be \(a_2<0\). If the initial speed of this phase is \(v_1\) and the final speed after deceleration is \(v_2\), the equation of the line is \(v=v_1 + a_2t\) (for the deceleration phase).

Step4: Analyze final constant - speed phase

After deceleration, when maintaining a new constant speed \(v = v_2\), the graph is again a horizontal line until she exits the highway.

Answer:

The graph has four distinct parts:

1. A straight line with a positive slope (acceleration).
2. A horizontal line (constant speed).
3. A straight line with a negative slope (deceleration).
4. A horizontal line (new constant speed until exit).