Differentiation of Vectors with Respect to Time - QS Study

Differentiation of vectors with respect to time:

If a vector quantity varies with time, then differentiation of that quantity is called differentiation of vector with respect to time.

For example, position vector r of a moving body depends on time t. Here vector r is a function of time, t

Explanation: Let consider that a bus is running on a plain and straight road from north to south. If the bus moves 2m in the first sec, 4m in the 2nd sec, 6 m in the 3rd sec; then we can say that the displacement x of the bus is always same. Now if a graph is drawn with displacement x along Y-axis and time t along X-axis, then it will be a straight line. (fig: a)

We can find ∆x and ∆t by drawing normal’s on X and Y axes from two points P and Q of this straight line. The ratio of ∆x and ∆t, i.e., slope ∆x/∆t will give the velocity of the bus. The slope at every point on this line will have same value i.e., velocity is the same. That means, the velocity for any interval of time (however small the interval may be) will be same. In this condition, the average velocity and instantaneous velocity of the bus is same.

But if the bus moves along a road having curvature or ups and downs and if the velocity of the bus needs to change frequently, then displacement x versus time t graph will not be a straight line. It will be a curve (Fig b). Here slope between any two points of the curve will not be same between other two points of the curve.

That means ∆x/∆t will be different for different points. Now if the time interval is considered exceedingly small, then rate of change of displacement, i.e., velocity will be almost equal to the actual velocity at that point. If ∆t tends to zero, then ∆x/∆t will be actual velocity.

i.e., limit∆t→0 ∆x/∆t = v; actual velocity

In calculus, limit∆t→0 ∆x/∆t is written as dx/dt

i.e., limit∆t→0 ∆x/∆t = dx/dt

It is the differentiation of the vector with respect to time. Here x is the displacement and d/dt is the operator.