One-Dimensional Motion – Application Of Calculus In Physics
Calculus is a very important branch of mathematics. In this branch, the main pillar is the infinitesimal magnitudes and a multitude of infinitesimal numbers. There is no better tool in mathematics than calculus to express any physical quantity [which is a quantitative property] in mathematical terms.
Modern calculus was developed in the 17th century by Issac Newton and Gottfried Wilhelm Leibniz independently. Calculus is a Latin word; it means ‘small pebble used in an abacus for counting’. The word calculus is also used in Latin as a synonym of counting.
In physics, it is important to know the relation among the variable quantities or how the change in one quantity affects another. There is no other way to analyse without the use of infinitesimal magnitudes and numbers. So, in physics, calculus is an indispensable tool.
Variable And Constant: A variable is a value that may change within the scope of the given problem or set of operations. A constant is a value that remains unchanged. Suppose, a greengrocer has a stock of 10 kg bitter gourd and he sells it at a price of Rs. 16 per kg.
If the seller does not change the price, it is constant. But the quantity of bitter gourd bought by individual buyers and its price are variables, because these may vary from 0 kg to 10 kg and from Rs. 0 to Rs. 160.
Real Variable And Complex Variable: A variable to which only real numbers are assigned as values is called a real variable. A variable which can take on the value of a complex number is called a complex variable.
Any complex variable has two parts—the real part and the imaginary part. Suppose z(=x+ iy) is a complex variable. For this variable, x and iy are the real and imaginary parts respectively. Here, x and y are real variables and i = √-1 is unit imaginary number or the imaginary unit.
Read and Learn More: Class 11 Physics Notes
We will mainly be using real variables in our discussion.
Function-Independent Variable And Dependent Variable: Any function relates to two variables or variable quantities. Of these, one is a dependent variable and the other, is an independent variable. Suppose a relation is expressed as y = f(x). We read the equation as y is a function of x. Here y and x are dependent and independent variables respectively. We generally express any functional relation as:
y = f(x) = ax²+ bx+ c or, y(x) = ax² + bx+ c
or, y = ax² + bx+ c [generally ‘(x) ’ is not written].
Here, if a, b and c are constants, then for any value of x, we can calculate the corresponding value of y.
Function: If we get only one value of a dependent variable y for a single value of independent variable x, then we can say y is a function of x. Calculus is based on such functions, y = x is a functional relation. But y² = x is not a functional relation. Actually, y² = x consists of two functions y = √x and y = -√x.
Differentiation: Suppose, y = f(x) is a functional relation, where x and y are respectively the independent and dependent variables. If x increases to x+ Δx i.e., if the increment of x is Δx, then y changes to y + Δy i.e., the increment of y is Δy.
So we can write it as an equation: Δy = f(x+Δx)- f(x)
Dividing both sides by Δx, we get \(\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}\)….(1)
= the change of y caused by a unit change of x.
Now, if Δx → 0, (i.e., the value of Δx tends towards zero or the value x is very small) we can write \(\frac{\Delta y}{\Delta x}\) as
⇒ \(\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\frac{d y}{d x}=f^{\prime}(x)\)….(2)
[we read \(\lim _{\Delta x \rightarrow 0}\) as : limit Δx tends to zero]
If Δx → 0, then the limiting value of \(\frac{\Delta y}{\Delta x}\) is expressed as \(\frac{dy}{dx}\) or f'(x). \(\frac{dy}{dx}\) or f'(x) is “the derivative of y with respect to x”. Actually, \(\frac{dy}{dx}\) is the rate of change of y with respect to x.
The process of determining the derivative is called differentiation. It must be remembered that \(\frac{dy}{dx}\) does not mean dividing dy by dx. It is only the symbol of the limiting process which is shown in equation (2).
It is to be mentioned that, when Δx → 0, the straight line A’B’ is the tangent to the curve y = f(x) at a point A and θ = θ’. Besides \(\frac{dy}{dx}\) or f'(x), we can also express the derivative of y with respect to x with the symbols — y’ or y1 or Dy or \(\frac{d}{dx}\)(y) or \(\frac{d}{dx}\){f(x)}.
The Meaning Of Δx → 0: The value of Δx tending to 0 means that the value of Δx is never exactly 0. Whatever value is close to zero we may imagine, the value of Δx will be even closer to zero. Suppose we imagine a value 0.00001 (or -0.00001) which is nearly 0. In that case, Δx can assume any value between 0.00001 to 0 (or -0.00001 to 0).
Slope: If we consider two points A(x1,y1) and B(x2,y2) on a straight line (line number 1) on a plane xy, then the slope of the straight line
m = \(\frac{y_2-y_1}{x_2-x_1}=\tan \theta\)
Now, instead of a straight line if we consider a curve (line number 2) then the slope is not equal at all the points on the curve. To measure the slope we need to take two points within a very small distance.
The curve between these two points is considered to be a part of a straight line. If the coordinates of these two points P and Q are (x, y) and (x+ dx, y+dy) respectively, then the slope of the curve at the point (x, y) is
m = \(\frac{(y+d y)-y}{(x+d x)-x}=\frac{d y}{d x}\)
So we can say, the slope of the curve on a plane xy at a point (x, y) = \(\frac{dy}{dx}\)
Derivatives Of Algebraic Functions
- \(\frac{d}{d x}\left(x^n\right)=n x^{n-1}\)
- \(\frac{d}{d x}\left(a x^n\right)=a n x^{n-1}\)
- \(\frac{d}{d x}\left(a^x\right)=a^x\) ln a [we can write \(\log _e a\) as ln a]
Derivatives Of Trigonometric Functions
- \(\frac{d}{d x}(\sin x)=\cos x\)
- \(\frac{d}{d x}(\cos x)=-\sin x\)
- \(\frac{d}{d x}(\tan x)=\sec ^2 x\)
- \(\frac{d}{d x}(\cot x)=-{cosec}^2 x\)
- \(\frac{d}{d x}(\sec x)=\sec x \tan x\)
- \(\frac{d}{d x}{cosec} x=-{cosec} x \cot x\)
- \(\frac{d}{d x}(\sin a x)=a \cos a x\)
- \(\frac{d}{d x}(\cos a x)=-a \sin a x\)
Derivatives Of Inverse Trigonometric Or Cyclometric Functions:
- \(\frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^2}}(|x|<1)\)
- \(\frac{d}{d x}\left(\cos ^{-1} x\right)=\frac{-1}{\sqrt{1-x^2}}(|x|<1)\)
- \(\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2}\)
- \(\frac{d}{d x}\left(\cot ^{-1} x\right)=\frac{-1}{1+x^2}\)
- \(\frac{d}{d x}\left(\sec ^{-1} x\right)=\frac{1}{x \sqrt{x^2-1}}(|x|>1)\)
- \(\frac{d}{d x}\left({cosec}^{-1} x\right)=\frac{-1}{x \sqrt{x^2-1}}(|x|>1)\)
Derivatives Of Logarithmic And Exponential Functions:
- \(\frac{d}{d x}\left(\log _e x\right)=\frac{1}{x}\)
- \(\frac{d}{d x}\left(\log _a x\right)=\frac{1}{x} \log _a e\)
- \(\frac{d}{d x}\left(e^x\right)=e^x\)
- \(\frac{d}{d x}\left(e^{a x}\right)=a e^{a x}\)
Basic Properties Of Differentiation:
Derivative Of A Constant: If f(x) = c (constant) \(\frac{d}{d x}\{f(x)\}=\frac{d c}{d x}=0\)
Derivative Of The Sum Or Difference Of Two Functions: If f(x) = g(x)±h(x) then, \(\frac{d}{d x}\{f(x)\}=\frac{d}{d x}\{g(x) \pm h(x)\}=\frac{d g}{d x} \pm \frac{d h}{d x}\)
Derivative Of The Sum Or Difference Of Two Functions Example:
y = \(\frac{6 x^6+8 x^2-2}{x^3} \text { or, } y=\frac{6 x^6}{x^3}+\frac{8 x^2}{x^3}-\frac{2}{x^3}\)
or, \(y=6 x^3+8 x^{-1}-2 x^{-3}\)
∴ \(\frac{d y}{d x}=\frac{d}{d x}\left(6 x^3\right)+\frac{d}{d x}\left(8 x^{-1}\right)-\frac{d}{d x}\left(2 x^{-3}\right)\)
= \(6 \times 3 x^{3-1}+8(-1) x^{-1-1}-2(-3) x^{-3-1}\)
= \(18 x^2-8 x^{-2}+6 x^{-4}\)
Derivative Of The Product Of Two Functions:
If f(x) = g(x)h(x) then, \(\frac{d}{d x}\{f(x)\}=\frac{d}{d x}\{g(x) h(x)\}=g \frac{d h}{d x}+h \frac{d g}{d x}\)
Derivative Of The Product Of Two Functions Example:
y = \((3 x-7)(5-6 x)\)
∴ \(\frac{d y}{d x}=(3 x-7) \frac{d}{d x}(5-6 x)+(5-6 x) \frac{d}{d x}(3 x-7)\)
= \((3 x-7)(-6)+(5-6 x)(3)\)
= \(-18 x+42+15-18 x=-36 x+57\)
Derivative Of The Ratio Of Two Functions:
If f(x)= \(\frac{g(x)}{h(x)}\) then, \(\frac{d}{d x}\{f(x)\}=\frac{d}{d x}\left\{\frac{g(x)}{h(x)}\right\}=\frac{h \frac{d g}{d x}-g \frac{d h}{d x}}{h^2}\)
Derivative Of The Ratio Of Two Functions Example:
y = \(\frac{(2 x+1)(3 x-1)}{x+5}=\frac{6 x^2+x-1}{x+5}\)
∴ \(\frac{d y}{d x}=\frac{(x+5) \frac{d}{d x}\left(6 x^2+x-1\right)-\left(6 x^2+x-1\right) \frac{d}{d x}(x+5)}{(x+5)^2}\)
= \(\frac{6 x^2+60 x+6}{(x+5)^2}=\frac{6\left(x^2+10 x+1\right)}{(x+5)^2}\)
= \(\frac{6 x^2+60 x+6}{(x+5)^2}=\frac{6\left(x^2+10 x+1\right)}{(x+5)^2}\)
Chain Rule Of Differentiation: If f(x) = x = g(z) then, \(\frac{d y}{d z}=\frac{d y}{d x} \cdot \frac{d x}{d z}\)…..(3)
Chain Rule Of Differentiation Example:
y = \(u^5 \text { and } u=x^2+3\)
∴ \(\frac{d y}{d u}=5 u^4=5\left(x^2+3\right)^4 \text { and } \frac{d u}{d x}=2 x\)
∴ \(\frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}=5\left(x^2+3\right)^4 \cdot 2 x\)
= \(10 x\left(x^2+3\right)^4\)
We can also express y = f(x) as x = g(y). In that case, if the value of \(\frac{dy}{dx}\) is nor 0, then from equation (3), we get
1 = \(\frac{d y}{d x} \cdot \frac{d x}{d y} \text { or, } \frac{d y}{d x}=\frac{1}{\frac{d x}{d y}}\)
Second Order Derivative: Second order derivative means, the derivative of the derivative of the function y = f(x) and it is written as \(\frac{d}{d x}\left(\frac{d y}{d x}\right) \text { or, } \frac{d^2 y}{d x^2}\)
Second Order Derivative Example:
x = \(3 \cos \pi t+4 \sin \pi t\)
∴ \(\frac{d x}{d t}=3(-\sin \pi t) \pi+4(\cos \pi t) \pi\)
= \(-3 \pi \sin \pi t+4 \pi \cos \pi t\)
∴ \(\frac{d^2 x}{d t^2}=-3 \pi(\cos \pi t) \pi+4 \pi(-\sin \pi t) \pi\)
= \(-\pi^2(3 \cos \pi t+4 \sin \pi t)=-\pi^2 x\)
Integration: Integration is the inverse process of differentiation. Suppose f(x) is a function of x and \(\frac{d}{dx}\){f(x)} = F(x)
i. e., the derivative of f(x) with respect to x is F(x), and F(x) is also a function of x.
It may be said that the integral of F(x) with respect to x is f(x) and it can be expressed by the equation
∫F(x)dx = f(x)…..(1)
F(x) is called the integrand. ‘ ∫ ’ and ‘dx’ are the symbols of integration.
Constant Of Integration: it is known, \(\frac{d}{d x}\left(x^5\right)=5 x^4\)
and \(\frac{d}{d x}\left(x^5+c\right)=5 x^4\) [as c is a constant]
So, the derivatives of functions x5 and x5 + c are the same. Then the integration of 5x4 should be written in general as x5 + c because here c is a constant and when c = 0, we get the function x5.
So, \(\int 5 x^4 d x=5 \cdot \frac{x^{4+1}}{4+1}+c=x^5+c\)
This constant c is called the integration constant. As this constant is indefinite, it is called the indefinite integration constant.
Definite Integral: For definite integral, equation number (1) can be written as \(\int_a^b F(x) d x=f(b)-f(a)\).
So, if the value of x changes from a to b, then the value of f(x) changes by f(b)-f(a). This [f(b) -f(a)] is called the definite integral of F(x) within the limits a and b. Here b is called the upper limit and a is called the lower limit.
The definite integral can be described as the area under the curve, \(\int_a^b f(x) d x\) is the area confined within the lines y = f(x), x-axis, x = a and x = b i.e., this area may be written as
area = \(\lim _{\Delta x_i \rightarrow 0} \sum_i f\left(x_i\right) \Delta x_i=\int_a^b f(x) d x\)
The symbol of integration i.e., ‘∫’ comes from the first letter of ‘summation1 and it is written as ‘long S’.
Basic Properties Of Integration
Integration Of The Product Of A Function And A Constant: If f(x) = ag(x) then ∫f(x)dx = ∫ag(x)dx = a∫g(x)dx
Integration Of The Sum Or Difference Of Two Functions: If f(x) = g(x)±h(x) then, ∫f(x)dx =∫{g(x)±h(x)}dx
=∫g(x)dx ± ∫h(x)dx
Integration Of The Product Of A Function And A Constant Example: \(f(x)=3 x^4-6 x^2+8 x-5\)
∴ \(\int f(x) d x=\int 3 x^4 d x-\int 6 x^2 d x+\int 8 x d x-\int 5 d x\)
= \(3 \times \frac{x^{4+1}}{4+1}-6 \times \frac{x^{2+1}}{2+1}+8 \times \frac{x^{1+1}}{1+1} -5 \times \frac{x^{0+1}}{0+1}+c\)
= \(\frac{3}{5} x^5-2 x^3+4 x^2-5 x+c\)
Interchange Of Upper Limit And Lower Limit Of A Definite Integral: \(\int_a^b f(x) d x=-\int_b^a f(x) d x\)
Interchange Of Upper Limit And Lower Limit Of A Definite Integral Example:
⇒\(\int_1^2 x^2 d x=\left[\frac{x^3}{3}\right]_1^2=\frac{8}{3}-\frac{1}{3}=\frac{7}{3}\)
But, \(\int_2^1 x^2 d x=\left[\frac{x^3}{3}\right]_2^1=\frac{1}{3}-\frac{8}{3}=-\frac{7}{3}\)
∴ \(\int_1^2 x^2 d x=-\int_2^1 x^2 d x\)
Insertion Of Any Intermediate Limit Between The Upper And Lower Limits: \(\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x\)
[c may be greater or less than both the upper and lower limits b and a]
Insertion Of Any Intermediate Limit Between The Upper And Lower Limits Example: \(\int_0^{\frac{\pi}{2}} \cos x d x=[\sin x]_0^{\pi / 2}=\sin \frac{\pi}{2}-\sin 0=1\)
Again \(\int_0^{\frac{\pi}{4}} \cos x d x+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos x d x\)
= \([\sin x]_0^{\pi / 4}+[\sin x]_{\pi / 4}^{\pi / 2}\)
= \(\sin \frac{\pi}{4}-\sin 0+\sin \frac{\pi}{2}-\sin \frac{\pi}{4}=1\)
∴ \(\int_0^{\frac{\pi}{2}} \cos x d x=\int_0^{\frac{\pi}{4}} \cos x d x+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos x d x\)
Integrals Of Algebraic Functions:
- \(\int x^n d x=\frac{x^{n+1}}{n+1}+c(n \neq-1)\)
- \(\int a x^n d x=a \int x^n d x=\frac{a x^{n+1}}{n+1}+c(n \neq-1)\)
- \(\int \frac{d x}{x}={m}|x|+c\)
- \(\int a^{m x} d x=\frac{a^{m x}}{m \ln a}+c \quad(a>0, a \neq 1)\)
Integrals Of Trigonometric Functions:
- \(\int \sin x d x=-\cos x+c\)
- \(\int \cos x d x=\sin x+c\)
- \(\int \tan x d x=\ln |\sec x|+c\)
- \(\int \cot x d x=\ln |\sin x|+c\)
- \(\int \sec x d x=\ln |\sec x+\tan x|+c=\ln \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right|+c\)
- \(\int {cosec} x d x=\ln |{cosec} x-\cot x|+c=\ln \left|\tan \frac{x}{2}\right|+c\)
- \(\int \sin m x d x=-\frac{\cos m x}{m}+c\)
- \(\int \cos m x d x=\frac{\sin m x}{m}+c\)
- \(\int \sec ^2 x d x=\tan x+c\)
- \(\int {cosec}^2 x d x=-\cot x+c\)
- \(\int \sec x \tan x d x=\sec x+c\)
- \(\int {cosec} x \cot x d x=-{cosec} x+c\)
Integrals Of Logarithmic And Exponential Functions:
- \(\int \ln a x d x=x(\ln a x-1)+c\)
- \(\int e^x d x=e^x+c\)
- \(\int e^{m x} d x=\frac{e^{m x}}{m}+c\)