Understanding Acceleration | The Derivative of Velocity and its Antiderivative

Acceleration is the antiderivative of…

Acceleration is the derivative of velocity

Acceleration is the derivative of velocity. In calculus, the derivative represents the rate of change of a function with respect to an independent variable. In the context of physics, velocity is the rate of change of displacement with respect to time. Therefore, if we take the derivative of velocity with respect to time, we obtain the acceleration.

Mathematically, if v(t) represents the velocity of an object at time t, then the derivative of v(t) with respect to time, denoted as a(t), gives the acceleration:

a(t) = dv(t)/dt

Here, dv(t) represents infinitesimal change in velocity and dt represents infinitesimal change in time. This expression represents the instantaneous rate of change of velocity with respect to time, or in other words, the acceleration of the object at any specific moment in time.

To find the antiderivative of acceleration, we would reverse this process. The antiderivative is the reverse of differentiation and allows us to find the original function when only the derivative is known. However, in this case, since acceleration is already the derivative of velocity, taking the antiderivative of acceleration would lead us back to velocity:

∫a(t) dt = v(t) + C

Here, ∫ represents the integral sign, a(t) is acceleration, and C represents the constant of integration. The antiderivative of acceleration is the original velocity function, up to an integration constant.

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