Important Derivations for Class 11 Physics Chapter 3 Motion in a Straight Line

Important Derivations for Class 11 Physics Chapter 3 Motion in a Straight Line

Here we are providing important derivations for Class 11 Physics Chapter 3 Motion in a Straight Line. Motion in a Straight Line or Motion in 1D derivations for class 11 Physics.

(1) Derivation of Kinematical Equations Using Calculus Method:

(A) First Kinematical Equation

Starting with the definition of acceleration: \[ a = \frac{{dv}}{{dt}} \] Performing separation of variables to get \(v\) and \(t\) on one side and \(u\) and \(a\) on the other side: \[ dv = a \, dt \] Integrating both sides from the initial velocity \(u\) at \(t = 0\) to the final velocity \(v\) at time \(t\): \[ \int_{u}^{v} dv = \int_{0}^{t} a \, dt \] Integrating the left-hand side with respect to \(v\) gives us: \[ v – u = \int_{0}^{t} a \, dt \] Now, evaluate the definite integral on the right-hand side: \[ v – u = \left[ at \right]_{0}^{t} \] \[ v – u = a \cdot t – a \cdot 0 \] \[ v – u = a \cdot t \] Finally, isolate \(v\) to get the first kinematical equation: \[ v = u + at \] This 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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