Instantaneous Rate of Change

In linear or circular motion, the instantaneous rate of change is related to the physical displacement of an object with respect to time. But the instantaneous rate of change is also related to the the movement of some quantity passing through a reference with respect to time.

Flow Rate

Flow rate usually refer to the volumetric flow rate of an object. Volume flow rate q is defined as the volume of the object passes through a given surface per unit time. When the given surface is equal to one unit, the velocity of the object flow is equal to the magnitude of the volume flow rate.

For a steady flow, the relationship between the final volume V, initial volume Vo, volume flow rate Q and time t can be expressed as a function:

Instantaneous flow rate can be defined as the instantaneous rate of change of the volume flow through a reference with respect to the time t. The derivative of the function is..

The volume flow rate is a constant for both a steady uniform flow and a steady nonuniform flow because the parameters of the flow do not change in time at every point. However only the uniform flow does have the parameters of the flow are the same for all spatial points along the flow, e.g. flowing through pipe with constant diameter. But for the nonuniform flow,  the parameters of the flow vary and are different at different spatial points along the flow, e.g. flowing through a tapering pipe.

For a steady flow with constant acceleration, the relationship between the final volume flow rate Q, initial volume flow rate Qo, acceleration a and time t can be expressed as a function:

Instantaneous acceleration can be defined as the instantaneous rate of change of the volume flow rate with respect to the time. The derivative of the function is.

Since V is a function of time f(t), the derivative Q is equal to the first derivative of function V with respect to time t.

The derivative a is equal to the derivative of function Q with respect to t. But Q is also a function of time, the derivative is

Therefore the derivative a is also equal to the second derivative of function V with respect to t.

Flow Rate Problem

A water supply system is operated by a water pump. The water supply after switching on the pump can be described by a volume function V with respect to time t.

Graphically

The information directly from the function is the total pumped water.

e.g. at t=1, the total pumped water is 3.1,

e.g. at t=2, the total pumped water is 8,

e.g. at t=3, the total pumped water is 13,

The first derivative can be used to determine the instantaneous volume flow rate. imply

Graphically

The information directly from the derivative is the instantaneous volume flow rate.

e.g. at t=1, the instantaneous volume flow rate is 4.59

e.g. at t=2, the instantaneous volume flow rate is 4.97.

e.g. at t=3, the instantaneous volume flow rate is 5

The second derivative can be used to determine the instantaneous acceleration of the instantaneous volume flow rate. imply

Graphically

The information directly from the derivative is the instantaneous acceleration.

e.g. at t=1, the instantaneous acceleration is 1.026

e.g. at t=2, the instantaneous acceleration is 0.804

e.g. at t=3, the instantaneous acceleration is 0.007