Instantaneous Rate of Change

Rate of change is the average change of function y=ƒ(x) between two instants, x and x+Δx. The derivative is the instantaneous rate of change by applying limit Δx to zero.

Linear Motion

For a linear motion with uniform speed v, the relationship between the travelled distance s, initial distance so, speed v and time t can be expressed as a function:

Instantaneous velocity can be defined as the instantaneous rate of change of the travelled distance with respect to the time of travelling. The derivative of the function s with respect to time t is.

For a linear motion with constant acceleration a, the relationship between the final speed v, initial speed vo, acceleration a and time t can be expressed as a function:

Instantaneous acceleration can be defined as the instantaneous rate of change of the travelling velocity with respect to the time of travelling. The derivative of the function v with respect to time t is.

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

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

 

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

Linear Motion Problem

For an object with a linear motion and the motion can be described by a distance function s with respect to time t.

Graphically

The information directly from the function is the travelled distance from the original location.

e.g. at t=1, the distance from the original location is 2,

e.g. at t=2, the distance from the original location is 2

e.g. at t=4, the distance from the original location is -4, the negative sign means the distance is measured from the opposite side of the original location relative to the original default measurement.

The first derivative can be used to determine the instantaneous verlocity of the object motion. imply

Graphically

The information directly from the derivative is the instantaneous velocity.

e.g. at t=1, the instantaneous velocity is 1

e.g. at t=2, the instantaneous velocity is -1, the negative sign means the direction of motion is opposite to the originial direction of the default motion.

e.g. In stationary state condition, the instantaneous velocity of the object is eqaul to zero. Imply

Therefore at t=1.5, the object remains stationary at its location with a zero instantaneous velocity. Imply the location is

The second derivative can be used to determine the instantaneous acceleration of the object motion. imply

The accelation is equal to -2, a negative acceleration. The value is a negative constant which imply it is a constant deceleration. And therefore the instantaneous velocity is decreasing with respect to time.  Graphically, the acceleration is