Question

given the following position vs time graphs, construct the corresponding velocity vs time and acceleration vs time graphs, create velocity and acceleration motion maps and describe the motion. if you see a dashed line in a graph it tells you the motion changes at that time.

Explanation:

Step1: Recall derivative relationships

Velocity is the derivative of position with respect to time ($v=\frac{dx}{dt}$), and acceleration is the derivative of velocity with respect to time ($a = \frac{dv}{dt}$).

Step2: Analyze left - hand side position - time graph

For the first part of the left - hand side position - time graph (before the dashed line), it is a straight - line with a positive slope. So the velocity is a positive constant. Since velocity is constant, acceleration is 0. After the dashed line, the position - time graph is a straight - line with a negative slope. So the velocity is a negative constant, and acceleration is still 0 as velocity is constant.

Step3: Analyze right - hand side position - time graph

The right - hand side position - time graph is a curve with an increasing slope. So the velocity is increasing (positive acceleration). The slope of the position - time graph starts from a non - zero value and keeps increasing, so the velocity starts from a non - zero value and increases with time, and the acceleration is positive and constant.

Step4: Motion description for left - hand side

The object moves with a constant positive velocity for the first part of the motion, then changes direction and moves with a constant negative velocity. There is no acceleration during either part of the motion.

Step5: Motion description for right - hand side

The object starts with a non - zero velocity and accelerates with a constant positive acceleration, so its speed (magnitude of velocity) is increasing over time.

Answer:

For the left - hand side: The object moves with a constant positive velocity initially, then changes direction and moves with a constant negative velocity. There is no acceleration. For the right - hand side: The object starts with a non - zero velocity and has a constant positive acceleration, so its velocity is increasing over time.