Question

Assume 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.

Do not want hand written answer and do not copy paste. Please type. Thanks.

Answer 1

The instantaneous velocity at any time can be obtained simply by differentiating the position function in time variable

If object moving along a straight line at time t then velocity will

Then acceleration will

Now we are using integral calculus, now calculating velocity function from acceleration function and the position function from the velocity function

Using fundamental thm of calculus

Now determining difference between the position at t₁  and the position at t₂

Remark If object is moving from position

x (t₁) to the position x (t₂)

Then is called displacement of object

Now totel distance between t₁ and t₂ is give by just integral of velocity with respect to time

I think now you understood, thanks