If you integrate acceleration with respect to time, you obtain the velocity of an object. This integration process is known as finding the antiderivative or integrating the acceleration function.
On the other hand, if you integrate velocity with respect to time, you obtain the displacement or position of an object. This integration process is known as finding the antiderivative or integrating the velocity function.
Mathematically, the integration of velocity can be expressed as:
displacement (or position) = ∫(velocity) dt
The integral of velocity over time gives you the change in position or displacement of an object. It represents the area under the velocity-time graph.
It's important to note that the integration of velocity gives you the position or displacement up to an arbitrary constant. This constant is usually determined by initial conditions or boundary conditions. In other words, if you integrate velocity to find the position, you need to specify the initial position or use additional information to determine the constant of integration and find the exact position at any given time.
In summary, integrating acceleration gives you velocity, and integrating velocity gives you displacement or position (up to a constant of integration).