Related Rates

For motion system with fixed relationship between its input and output, the derivatives, instantaneous rates of change can also be used as related rates of change for determining the unknown rate of change by the known rate of change.

Related Motion

The most common related motions are the physical related motions in a mechanical system. Sometimes the related motion can also be other physical systems, for example optical projection.

Related Linear Motion - Falling Ladder Problem

A 25 long ladder is leaning against a wall. When the bottom of ladder is moved outward at x=3 with constant speed Vx=3, the top of ladder is being moved downward simultaneously.  Since the ladder is rigid and fixed in length, the instantaneous vertical velocity Vy at time 7 is:

 

According to the given information. The unknown rate of change is the linear displacement along y direction with respect to time t and the known rate of change is the linear displacement along x direction with respect to time t. Both x and y are the Cartesian coordinates. The relationship between x and y in Cartesian coordinates can be related by the length of the ladder through the Pythagorean theorem, i.e. L2=x2+y2. Therefore the mathematical model of the falling ladder is:

The relationship of the two cartesian instantaneous velocity vx and vy are:

Therefore the instantaneous velocity vy is a function of x and y. And y can be expressed as:

And the position of the bottom of ladder at time t is:

Subsitute all variables and get:

Graphically:

Intantaneous velocities plot against time:

Related Linear Motion - Moving Rod Problem

A guided rod is moving along the x and y axis with fixed enclosed area A=36. When the guided rod is moved outward along x-axis at x=3 with constant speed Vx=3, the guided rod is being moved downward simultaneously.  Since the guided rod is rigid and the enclosed area is fixed, the instantaneous vertical velocity Vy along y-axis at time 7 is:

According to the given information. The unknown rate of change is the linear displacement along y direction with respect to time t and the known rate of change is the linear displacement along x direction with respect to time t. Both x and y are the Cartesian coordinates. The relationship between x and y in Cartesian coordinates can be related by the enclosed area, i.e. A=xy/2. Therefore the mathematical model of the falling ladder is:

The relationship of the two cartesian instantaneous velocity vx and vy are:

And the position of the bottom of ladder at time t is:

Subsitute all variables and get:

Graphically:

Intantaneous velocities and displacements plot against time: