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Important Derivations for Motion in a Straight Line | Physics for JEE Main & Advanced PDF Download

Important Derivation

1. Derivation of Kinematical Equations Using Calculus Method:

(A) First Kinematical Equation
Starting with the definition of acceleration:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & Advanced

Performing separation of variables to get v and t on one side and u and a on the other side:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & Advanced

Integrating both sides from the initial velocity u at t=0 to the final velocity v at time t:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & AdvancedIntegrating the left-hand side with respect to v gives us:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & AdvancedNow, evaluate the definite integral on the right-hand side:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & AdvancedFinally, isolate v to get the first kinematical equation:
Important Derivations for Motion in a Straight Line | Physics for JEE Main & AdvancedThis is the first kinematical equation derived using the calculus method, which relates the final velocity v of an object to its initial velocity u, acceleration a, and time t.

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