Estimating the distance traveled by a vehicle using its recorded velocity over time is a common problem in physics and engineering. When velocity data is available at discrete time intervals, rather than as a continuous function, numerical integration methods such as the trapezoidal rule are often employed to approximate the total displacement.
The trapezoidal rule works by dividing the total time interval into several equal segments. Within each segment, the recorded velocities at the endpoints are connected with a straight line, forming a trapezoid on a velocity–time graph. Each trapezoid has a vertical side representing the velocity at the beginning and end of the segment, and a horizontal base corresponding to the time interval.
To approximate the displacement, the area of each trapezoid is calculated. This area represents the average of the velocities at the endpoints of the segment, scaled by the duration of the segment. Summing the areas of all trapezoids yields an estimate of the total distance traveled over the entire time period.
In this process, the velocities recorded at the beginning and end of the total time interval are used only once, while the velocities at the intermediate points contribute to two adjacent trapezoids. This systematic approach leads to a reliable approximation of displacement, especially when the number of intervals is sufficiently large. The trapezoidal rule is therefore a fundamental technique in numerical analysis, providing a practical way to integrate empirical data from motion or other time-dependent processes.
Imagine a truck moving along a highway. To find the distance it travels during a given time interval t, find the area under its velocity–time graph.
Since the speed of the truck is known at discrete time points, this area can not be calculated precisely. Instead, the area is approximated using the trapezoidal rule. To do this, divide the total time interval into n equal subintervals, each with a width of Δt.
In this method, the two endpoints of each subinterval are connected by a straight line, forming trapezoids.
For a given subinterval, which has two vertical sides representing the velocities and a horizontal base representing the time width, the area of this trapezoid is the average of the lengths of its two vertical sides multiplied by the horizontal distance.
Adding the areas of all n trapezoids gives an approximation of the displacement.
After algebraic manipulations and solving this summation, the first and last velocities appear once, while interior velocities appear twice because each belongs to two trapezoids.
After combining like terms, the resulting expression approximates the total displacement over the specified time interval.
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