Question

What do slope values from the tangents of position-time graphs represent?

Answer 1

The slope is instantaneous speed.

Explanation:

This has been extensively edited to avoid the appearance, that I admit my explanation gave, that slope yielded average speed.

Let me start by describing how I imagine that you would determine the slope of the curve at a point. Draw a tangent to the curve at some point and extend that tangent so that it intersects the position axis. From the point of contact between the curve and the tangent, drop a line vertically down to the position axis. You now have a triangle formed by the tangent, the vertical line, and the position axis.

Considering the angle between the tangent and the position axis, the opposite side represents time and the adjacent side represents distance. The slope then is opposite/adjacent, which would have dimensions of distance/time.

Therefore the fraction that yields slope also yields instantaneous speed at that point.

I hope this helps,
Steve