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.
Table of Contents
(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 \(t\).Related Posts
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