AI Tutor
If a GPS navigation app tells you that your average speed over a two-hour drive was 50Β mph50\text{ mph}, does that mean your speedometer read 50Β mph50\text{ mph} the whole time?
You might have waited at a red light at 0Β mph0\text{ mph} and cruised down the highway at 70Β mph70\text{ mph}. To understand how motion changes from moment to moment, we use a displacement-time graph.
In physics, displacement (ss) is an object's straight-line position relative to a reference origin, while rectilinear motion simply means moving along a single straight line.
How do we extract an overall measure of speed across an entire journey from a curved graph?

Chords and Average Velocity

The slope or gradient of any line is defined as the vertical change (Ξ”s\Delta s) divided by the horizontal change (Ξ”t\Delta t). A straight line connecting any two points on a curve is called a chord or secant line.
The gradient of a chord on a displacement-time graph gives the average velocity (vavgv_{\text{avg}}) over that time interval: vavg=Ξ”sΞ”t=s2βˆ’s1t2βˆ’t1v_{\text{avg}} = \frac{\Delta s}{\Delta t} = \frac{s_2 - s_1}{t_2 - t_1}
Let's work through an example: a drone's displacement is modeled by s(t)=t2+2ts(t) = t^2 + 2t meters. At t1=1Β st_1 = 1\text{ s}, its position is s1=12+2(1)=3Β ms_1 = 1^2 + 2(1) = 3\text{ m}, and at t2=4Β st_2 = 4\text{ s}, its position is s2=42+2(4)=24Β ms_2 = 4^2 + 2(4) = 24\text{ m}.
The chord gradient gives the average velocity: vavg=24βˆ’34βˆ’1=213=7Β m/sv_{\text{avg}} = \frac{24 - 3}{4 - 1} = \frac{21}{3} = 7\text{ m/s}
Average velocity covers a span of time, but what if you need to know the exact speed at the precise instant t=3Β st = 3\text{ s}?

Tangents and Instantaneous Velocity