Speed of a car The accompanying figure shows the...

Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill. a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure $2.6 .$ What are the appropriate units for these slopes? b. Then estimate the car's speed at time $t=20 \mathrm{sec}$ .

Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill.
a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure $2.6 .$ What are the appropriate units for these slopes?
b. Then estimate the car's speed at time $t=20 \mathrm{sec}$ .

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Key Concepts

Slope of a Secant

In a distance-time graph, the slope of a secant line connecting two points represents the average rate of change of distance with respect to time over that interval. This calculation gives the car's average speed between the two time points.

Average Speed

Average speed is determined by dividing the total change in distance by the total time elapsed. In graphical terms, it corresponds to the slope of the secant between two points on the distance versus time plot.

Instantaneous Speed (Derivative)

Instantaneous speed is the car's speed at a specific moment, which is conceptually obtained as the limit of the average speeds over smaller and smaller time intervals. Graphically, this is approximated by the slope of the tangent line at the point of interest on the distance-time curve.

Units of Velocity

When distance and time are measured in units such as meters and seconds respectively, the appropriate units for velocity (or speed) are derived as distance per time, for example, meters per second (m/s). This unit reflects how many meters are covered in one second.

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