2016 Jan

Physics and Differential Calculus

January 7, 2016

Physics and Differential Calculus

Mathematics and physics are inseparably entangled. The role of mathematics in solving problems of physics is well known. The challenge mathematicians have faced in solving some of the problems of physics deserves a mention. Calculus is one such mathematical tool without which physics would be lagging behind by hundreds of years. Weather prediction, space exploration, prediction of the behavior of complex reaction networks in living organisms, power systems are a very few of plethora of things that are unimaginable without calculus. Let’s first discuss differential calculus and its role in physics.

Let’s say an object moves 10 meters in 10 seconds of time, the velocity of that object is defined by 1 meter per second. Given that the same object travels 20 meters in next 10 seconds, then the velocity of the object is 2 meters per second. And if the object moves 30 meters in next 10 seconds, the velocity is 3 meters per second. Imagine a graph with time on X-axis and displacement on Y-axis. When the above information is plotted on the graph, one can see three straight lines with different slopes. Here the object has a velocity of 1 m/s for the first 10 meters followed by 2 m/s and 3 m/s for the next two 10 second intervals respectively. The velocity of the object changed every 10 seconds.

Now imagine a situation where the velocity of the object changes every 5 seconds. Then the velocity changes more frequently. If you compress it further, let’s say, to the point that velocity changes instantly then the graph looks like a smooth curve. Velocity of such a system changes instantly. The velocity of such a system at every point can be obtained by drawing a tangent at that point and the slope of the tangent is equal to the velocity at that instant. Let’s say position of a system can be expressed as a function of time f(t) (for example something like t2+2 i.e. position at time t is t2+2). The velocity of such a system changes at every instant.

How do we find the velocity of the system at every instant? One can assume a curve to be collection of straight lines of different slopes and of infinitesimally small length. Just like the case our earth that is round on a whole but small parts of it are flat. Now from the function one can take very small piece of time and assume that the curve remains like a straight line there. The slope of that line is the velocity in that small interval of time. We have also learnt in our Physics tuition classes that the area under the line is the displacement. Using the concept of limits one can push the size of time interval to infinitesimally small length that gives us slope at that instant of time. The slope at that instant of time is velocity at that instant. Differentiation is nothing but finding slope at every instant. Thus when a function of time is given and you differentiate it you get slope at every instant. In our example when the position function ‘t2+2’ is differentiated we get another function given by ‘t’ that means that velocity at a time 10 units is 10 units.

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