Most of you already know about the terminologies of velocity and acceleration (see Kinematics III – speed, velocity and acceleration if you need to clear basics)). In this article we move from basic concepts to a calculus based view of Velocity and Acceleration. Let’s discuss about velocity first
Velocity and introduction to Derivatives:
The rate of change of displacement of a body is known as velocity. It is a vector quantity which has both magnitude and direction. Velocity can be positive, 0 and negative as well Velocity =displacement/time
The real question arises when someone asks you to find the velocity or speed of the body at a particular instant of time. Like for example what was the speed with which you were travelling at exactly the 4th sec? To answer such questions we have instantaneous velocity/speed. Instantaneous speed means the velocity of the particle at a particular time. To find this instantaneous speed we use derivatives. Now what does it mean to take derivatives? lets see…
If you lack knowledge in basic Calculus then read the next part or skip to “velocity using derivatives”
Short Intro to Derivatives
For example, look at the graph below.
The above diagram is a graph of a quadratic function. We choose two points on the graph and draw a line between them (in this fig, the points are shown). The line or more accurately the slope (steepness) of the line gives us the average change of the value of the function with respect to x. If this was a displacement-time graph then the slope of the red line would give us the average velocity.
The formula for slope is => (y2 – y1)/(x2 – y2)
But this is not derivatives (yet). Now imagine bringing the first point (0.4) close to point (3,-1) . bring it closer and closer and closer such that the two points Almost coincide. the line would appear like this-
This line appears to be touching the graph of the function at exactly one point. This line will always be a tangent line and the slope of this line will give the INSTANTANEOUS speed of an object. In our case we get the instantaneous velocity.
You can take the derivative of a function as many times as you like. derivatives are usually represented in two ways. Either as dy/dx or f'(x). the ” ‘ ” represents derivative of function f in second example. if you take two derivatives then you represent it like this- F”(x). And for 3 derivatives, f”'(x).
Integration is the opposite of derivative. it is sometimes also called the Anti-derivative. For this article all you need to know is that the integration of the derivative of the function is the function itself. Meaning if you integrate f'(x) then you get f(x)
Velocity using derivatives
The instantaneous velocity is given by:
V(ins)= dr/dt which is obtained by the first order derivative of displacement with respect to time .
As V=dr/dt it also implies dr= vdt,
Now integrating on LHS from a limit say initial displacement r1 to final displacement r2 and on RHS from 0- t we get r2-r2 = vt. this can be written as v = (r2-r1)/t that is change in position of body with respect to time. When we plot the graph of displacement versus time then the slope of the graph gives us the velocity.
Example: The function of position of body is given by- y = t^3 – t. find velocity at t=3.
solution: take derivative of y with respect to t. y’ = 3t^2 – 1. substitute t=3. answer is 3(9) – 1 = 26 metres/second.
Finding maximum and minimum velocity
To determine the maximum and minimum velocity of a object we need to use the concept of local maxima and local minima which will give us the maximum and minimum velocity in a particular time interval. To find maximum velocity find the first derivative of the velocity or the second derivative of the position and equate it to 0. in this way you will get some values of x. Then find the second derivative of velocity to determine whether the point is a local maxima or minima. Then substitute the values of x which you have found in the function of velocity.