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.

Please Type and Include a Figure. Do Not Copy Paste Any Answer

Answer 1

Instantaneous velocity is dx/dt. Now dx is small change in distance whereas dt is small change in time. We know that velocity is displacement/time. Now, it a very small time, the vehicle doesn't change its direction. Therefore we can consider it to be displacement. Since time is very very small, therefore displacement is very very small. Hence it is given by dx.

Taking time to be very very small is equivalent to talking about an instance. Thus velocity is known as instantaneous velocity. Now, we have to figure out what integration of velocity with respect to time means.

Integration with respect to time gives area under the curve of velocity with respect to time. We can divide the area into summation of rectangles with width dt and length v(t_i​​​​​​) where dt=t_i-t_i-1. Then, each such rectangule gives approximate distance covered in that time duration using the fact that distance =velocityxtime. Thus area under the curve gives total distance covered. And thus integral with respect to time gives total distance covered.