Exercise 7.1 Class 12 Math Solutions - Unit 7 Kinematics, Punjab Board
Step-by-step solutions to Exercise 7.1, Unit 7: Kinematics, for Class 12 Punjab Board students. Covers rectilinear motion, scalar vs vector kinematic quantities, average and instantaneous velocity, displacement-time (s-t) and velocity-time (v-t) graph analysis, equations of motion under constant acceleration, and vertical free fall and upward projection problems, worked out question by question.
Exercise Overview
Exercise 7.1 initiates Unit 7: Kinematics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum by analyzing rectilinear motion-the motion of particles along a one-dimensional straight-line trajectory. The exercise formally distinguishes scalar physical quantities (distance traveled, average speed (v_{\text{avg}} = \frac{\text{total distance}}{\text{total time}})) from vector quantities (displacement (\vec{s}), average velocity (\vec{v}_{\text{avg}} = \frac{\Delta \vec{s}}{\Delta t}), and instantaneous velocity (\vec{v} = \frac{d\vec{s}}{dt})). Instantaneous acceleration is defined as the first time derivative of velocity and the second time derivative of position: (\vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{s}}{dt^2} = v\frac{dv}{ds}).
The exercise thoroughly develops graphical kinematic interpretations: on a displacement-time ((s)-(t)) graph, the slope of the tangent at any point yields the instantaneous velocity ((\frac{ds}{dt})); on a velocity-time ((v)-(t)) graph, the gradient represents instantaneous acceleration ((\frac{dv}{dt})), while the definite integral or area under the curve represents the net displacement ((\int v \, dt)). For uniform motion with constant acceleration (a), the standard kinematic equations are derived and applied: (v = u + at), (s = ut + \frac{1}{2}at^2), and (v^2 - u^2 = 2as). Vertical motion under gravity ((a = -g)) is investigated for free fall from rest ((u = 0 \implies v = \sqrt{2gh})) and vertical upward projection ((H_{\max} = \frac{u^2}{2g}), total time of flight (T = \frac{2u}{g})).
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 7.1 is a fundamental source of Section B Short Questions (graph gradient calculations, velocity/acceleration differentiations, stopping distance problems) and Section C Long Questions (multi-stage rectilinear journeys and vertical motion systems under gravity). These step-by-step solutions provide rigorous mathematical calculus derivations, clear coordinate conventions, and exact algebraic evaluations for every textbook problem.