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Velocity is calculated from the area under an acceleration-time curve. For non-constant acceleration, divide the area into smaller rectangles with width Δt and height equal to average acceleration. Multiplying Δt by average acceleration gives the change in velocity. Sum all rectangle areas to find total velocity change between two time points.

The area under a velocity-time graph represents displacement or position change. Similar to acceleration-time graphs, the area is divided into rectangles with width Δt and height equal to average velocity. The product of Δt and average velocity gives displacement for each interval. Summing all rectangle areas yields total displacement between two time points.

When Δt approaches zero, average acceleration approaches instantaneous acceleration, and average velocity approaches instantaneous velocity. The sum of rectangle areas transitions from a discrete approximation to a continuous mathematical representation. This continuous form is expressed as an integral of instantaneous acceleration or velocity over the time interval.

The product of acceleration and time equals the change in velocity. Add this change to the initial velocity to find final velocity. For example, a cyclist with initial velocity 11.5 m/s accelerating at 0.500 m/s² for 7.00 s reaches a final velocity of 15.0 m/s using this relationship between acceleration, time, and velocity change.

The graphical method uses area calculations under acceleration-time and velocity-time curves. Divide the area under each curve into rectangles, calculate each rectangle's area, and sum them to find total velocity change or displacement. This approach works for both constant and non-constant acceleration by approximating curves with rectangular sections.

An inertial navigation system tracks an aircraft's acceleration and uses it to calculate velocity and position throughout flight. The pilot inputs the initial position and velocity before takeoff. The system then applies the graphical or integral method to acceleration data, continuously updating the aircraft's velocity and position estimates during flight.

Average acceleration or velocity represents the height of each rectangle in graphical analysis. As Δt decreases, these average values increasingly approximate instantaneous values at specific time points. When Δt approaches zero, the rectangular approximation becomes exact, and the sum of areas transforms into an integral of instantaneous acceleration or velocity.