3.13:
Velocity and Position by Graphical Method
Velocity and displacement can be calculated from the area under the curve for the acceleration-time and velocity-time graphs, respectively.
For non-constant acceleration, the area under the acceleration-time curve is split into smaller rectangles of width, Δt, and height, average acceleration. Δt times average acceleration is the change in velocity.
The sum of areas of rectangles gives the total change in velocity during t1 to t2.
When Δt approaches zero, the average acceleration approaches instantaneous acceleration, and the sum can be replaced with an integral. The area is then represented as the integral of instantaneous acceleration.
Similarly, the area under the velocity-time curve is split into smaller rectangles of width, Δt, and height, average velocity. Delta t times average velocity is the change in position or the displacement. The total displacement during t1 to t2 is equal to the sum of the areas of all rectangles.
When Δt approaches zero, the average velocity approaches instantaneous velocity, the area is represented as the integral of instantaneous velocity from time t1 to t2.
3.13:
Velocity and Position by Graphical Method
Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph and the velocity-time graph gives the change in velocity and position, respectively. In the case of an airplane, its acceleration is tracked using the inertial navigation system. The pilot provides the input of the airplane's initial position and velocity before takeoff. The inertial navigation system then uses the acceleration data to calculate the airplane's position and velocity throughout the flight.
For non-constant acceleration, the area under the acceleration-time curve is split into smaller rectangles, where the width is Δt, and the height is average acceleration. The quantity Δt multiplied by average acceleration is the change in velocity. Thus, the sum of the areas of the rectangles gives the total change in velocity from t1 to t2. When Δt approaches zero, the average acceleration approaches instantaneous acceleration, and the sum can be replaced with an integral. The area is then represented as the integral of instantaneous acceleration. Similarly, the displacement can be calculated from the area under the velocity-time curve.
Consider another example. A cyclist sprints at the end of a race to clinch a victory. They have an initial velocity of 11.5 m/s, and accelerate at a rate of 0.500 m/s2 for 7.00 s. What is their final velocity?
The known quantities are initial velocity (11.5 m/s), acceleration (0.500 m/s2), and time (7.00 s). The product of acceleration and time is equal to the difference between final and initial velocity. Using the values of initial velocity, acceleration, and time, the final velocity is calculated to be 15.0 m/s.
This text is adapted from Openstax, University Physics Volume 1, Section 3.6: Finding Velocity and Displacement from Acceleration.