The process of finding the derivative or the differential coefficient of a function w.r.t. the variable, on which it depends is called differentiation. Integration is the process of finding the function, whose derivative is given. for this reason, the process of integration is called the inverse process of differentiation.
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In this article, we will cover the mathematical tools used in kinematics. These tools are crucial for studying any chapter, especially in the mechanics section, where a solid understanding of basic mathematics is essential for solving problems.
So let’s read the entire article to know in-depth about the differentiation and integration to use in kinematics to make the calculation easier.
Differentiation is very useful when we have to find rates of change of one quantity compared to another.
If y is one quantity and we have to find the rate of change of y with respect to x which is another quantity
For a y V/s x graph
We can find the slope of the graph using differentiation
I.e Slope of
- Some important Formulas of differentiation
-
Example-
Similarly
The opposite process of differentiation is known as integration.
Let x , and y be two quantities, using differentiation we can find the rate of change of y with respect to x , Which is given by
But using integration we can get the direct relationship between quantities x and y
So let
Or we can write
Now integrating on both sides we get the direct relationship between x and y
Where
For a y V/s x graph
We can find the area of the graph using the integration
Example 1. Displacement of a particle as a function of time is given as
1) Independent of
2) 3at
3)
4)
Solution:
The displacement of a particle as a function of time is given as
The rate of Displacement-
Hence, the answer is option (1).
Example 2. If
1)
2)
3) 3
4)
Solution:
The formula of differentiation
Given -
So using the formula we get
Hence, the answer is option (1).
Example 3. Then what is the relation between
1)
2)
3)
4)
Solution:
The formula of integration-
Given-
As at
so,
Hence, the answer is option (1).
Example 4. The distance travelled by an accelerated particle of mass
1) 18
2) 12
3) 6
4) 24
Solution:
As we learned
Some important Formulas of differentiation
So
now using the formula
Hence, the answer is option (1).
Example 5. If
1)
2)
3)
4)
Solution:
Hence, the answer is the option (3).
The derivative or differential coefficient of a function is the limit to which the ratio of the small increment in the function to the corresponding small increment in the variable (on which it depends) tends to, when the small increment in the variable approaches zero. The concept of integration is used in physics to make measurements when a physical quantity varies in continuous manner.
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