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I'm a fresh Calculus $1$ student struggling to make this graph somehow, I've been studying Chapter $1$ from James Stewart's single variable Calculus early transcendentals. Excercise $1.1$ is easy up until this question:

An airplane takes off from an airport and land an hour later at another airport, $400$ miles away. If $t$ represents the time in minutes since the plane has left the terminal building, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane: Sketch a possible graph of the vertical velocity.

This site draws the graph but I can't understand this, why is this so? Why the graph flips? I'm terrible when physics comes in Math, any formula, I guess $v_0\sin\theta - gt$, would be helpful, in terms of explaining the graph. If this graph is correct, I'm looking for an explanation of the sketch.

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You do not need maths. Remember the vertical velocity is zero in level flight. As the plane takes off, it's vertical velocity increases from zero to a constant value. Then as the plane levels off, the vertical velocity decreases. The key thing to remember for the second half of the picture is that velocity is a vector. So as the altitude decreases the velocity is a negative value.

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  • $\begingroup$ And in principle you can draw a lot of them making the scenario more or less realistic. The airplane can level and cruise or unrealistically ascend and immediately descent etc $\endgroup$ – Alchimista Jan 9 '19 at 9:24

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