Question

From indefinite integration of ??/?? = ?,  ? is constant, find the velocity at time ?. Explain the meaning of the integration constant.

From the indefinite integration of the answer above, find the position of the object at time ?.

Explain the meaning of the second integration constant.

Answer 1

The integration of the acceleration function with respect to time gives the velocity function

Here, the velocity is at a time . The constant represent the initial velocity of the body, i.e the velocity at time .

We can represent the velocity as the instantaneous change of position with respect to time.

Here, we can see the position of the particle at any time is the sum of the distance travelled during its acceleration phase(think of the term as the area underneath a velocity-time graph that looks like a right triangle), the product of the initial velocity and the time and the constant

is hence the initial position of the particle, i.e the position of the particle at .