Question

The reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation.

Please type and include figure to explain.  do Not Copy Paste Any Answer

Answer 1

Assume you are traveling from point A to point B, Then the average velocity during the trip from a to be is by definition given by,

In mathematical motation this thing can be intrepreted as

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time , we are at A. If the time elapsed to get to B is , then we will be at B at time

If is the distance from A to B, then the average velocity is

The instantaneous velocity (at A) will be found when get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have

If describes the position at time t, then . In this case, we have

In other words the derivative of the displacement function at any point in the domain w.r.t time is simply equals to the velocity and is written as

This is the Physical Significance of derivative at a point.

Physical meaning of integral of the velocity,

We know that the integration is the reverse process of the derivative also thus called as anti derivative it means if we integrate the above equation of velocity w.r.t t between bounds we end up with

But by the definition of definite integral it is area under the velocity curve. hence the displacement.

i.e Displacement is an integral of velocity over time.