The equations of motion are derived from the indefinite integral of acceleration.verify
To verify that the equations of motion are derived from the indefinite integral of acceleration, let's start by understanding the concept of indefinite integration.
Indefinite integration represents the process of finding the antiderivative (or integral) of a function without the need for specific limits. In other words, it helps us to find a function whose derivative is equal to the given function.
In the case of motion, let's consider the one-dimensional motion of a mass under the influence of a force. Newton's second law of motion states that the sum of all forces acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, it can be written as:
F = m * a
Where F is the net force, m is the mass, and a is the acceleration.
Now, if we isolate the acceleration term, we get:
a = F/m
To proceed, we can integrate both sides of this equation with respect to time, t:
∫ a dt = ∫ (F/m) dt
The integral on the left side represents the indefinite integral of acceleration with respect to time, while the integral on the right side represents the indefinite integral of force divided by mass with respect to time.
Integrating the left side will give us a function that represents the velocity, v(t), and integrating the right side will give us a function that represents the displacement, x(t). Therefore, the equation becomes:
∫ a dt = ∫ (F/m) dt
v(t) = ∫ (F/m) dt
x(t) = ∫ v(t) dt = ∫ ∫ (F/m) dt dt
By applying the process of indefinite integration, we obtain the equations of motion:
v(t) = ∫ (F/m) dt
x(t) = ∫ v(t) dt = ∫ ∫ (F/m) dt dt
These equations relate the acceleration, velocity, and displacement of an object based on the net force exerted on it. Hence, we have verified that the equations of motion can indeed be derived from the indefinite integral of acceleration.