Final answer:
From a car's position vs. time graph, one can determine the corresponding velocity vs. time and acceleration vs. time graphs, utilizing the slopes for instantaneous velocity and acceleration.
Step-by-step explanation:
A car rolls down an incline starting from rest and a position vs. time graph of this motion is made. From this graph, one can derive both the velocity vs. time and acceleration vs. time graphs by understanding the relationships between position, velocity, and acceleration. The slope of the position vs. time graph provides the velocity of the car at any given moment, and the slope of the velocity vs. time graph provides the acceleration.
When looking at a position vs. time graph such as Figure 2.13, which shows a jet-powered car speeding up, we note that the instantaneous velocity is the slope of the tangent at any given point. If the position vs. time graph shows a curved line where the car is speeding up, the corresponding velocity vs. time graph will start from zero (since the car starts from rest) and increase as the car accelerates, reflecting the slope of the position-time graph.
If the slope of the position-time graph levels out, indicating a constant velocity, then the velocity-time graph will become horizontal, showing that the acceleration has reduced to zero. A return trip to the starting position would result in the graph returning back to the origin, displaying symmetry if the return speed is the same as the outward journey.