Instead of graphing the arc length against the height, Olivia decides to make graphs of Paul's x - and y - coordinates against time. Suppose paul walks 1 meter per second. How would Olivia's graph compare to her graph of arc length against the height?
Is this a follow-up to a previous problem? Not enough info here.
If Paul walks at a constant speed of 1 meter per second, Olivia's graph of Paul's x-coordinates against time would be a line with a positive slope, representing the distance covered in the x-direction over time.
On the other hand, Olivia's graph of Paul's y-coordinates against time would also be a line, but with a non-linear profile due to the parabolic nature of the projectile motion. The slope of this graph will vary as Paul's height changes with time.
In comparison to Olivia's graph of arc length against the height, her graph of the x and y coordinates against time would provide a more detailed representation of Paul's motion, showing the changes in both horizontal and vertical positions with respect to time.
To compare Olivia's graph of Paul's x- and y-coordinates against time with her graph of arc length against height, we first need to understand the relationship between them.
In the case of arc length versus height, Olivia is graphing the distance Paul has traveled along the curve (arc length) against the height of the curve. This would show how the height of the curve changes as Paul moves along it.
On the other hand, if Olivia graphs Paul's x- and y-coordinates against time, she would be showing the position of Paul in terms of his x- and y-coordinates at different points in time. With Paul walking at a constant speed of 1 meter per second, the x-coordinate would increase by 1 unit each second, while the y-coordinate would remain constant (assuming Paul is walking in a straight line).
Therefore, Olivia's graph of Paul's x-coordinate against time would be a straight line with a positive slope of 1. The y-coordinate graph would be a horizontal line since it does not change.
Comparing this to the arc length versus height graph, Olivia's graph of Paul's coordinates against time would be linear, while the arc length against height graph would likely be non-linear, following the shape of the curve.
In summary, Olivia's graph of Paul's x- and y-coordinates against time would show a linear relationship, while her graph of arc length against height would represent the non-linear nature of the curve.