1. Three Fundamental Equations of Uniformly Accelerated Motion

For motion in a straight line with constant acceleration a, initial velocity u, final velocity v, displacement s, and elapsed time t: (1) First equation (velocity-time relation): v = u + at; (2) Second equation (displacement-time relation): s = ut + (1/2)at^2; (3) Third equation (velocity-displacement relation): v^2 = u^2 + 2as. An essential fourth relation connecting average velocity: s = [(u + v)/2] · t. Note: These equations are valid strictly when acceleration is uniform (constant in magnitude and direction). If acceleration varies with time or position, calculus integration must be used: v = ∫a dt and s = ∫v dt.

2. Distance Covered in the nth Second

To calculate the distance traveled specifically during the nth single second (from t = n-1 to t = n), subtract the displacement at (n-1) seconds from the displacement at n seconds: s_n = s(n) - s(n-1) = u + (a/2)(2n - 1). Common exam trap: Notice that n has the dimension of time [T], so the factor (2n - 1) carries dimension [T], making the entire term dimensionally consistent with displacement [L] despite appearing arithmetic.

3. Vertical Motion Under Gravity (Free Fall)

Setting the upward direction as positive (+y) with acceleration due to gravity acting downwards (a = -g = -9.8 m/s^2 or -10 m/s^2): (1) Body dropped from rest (u = 0): v = -gt, h = -(1/2)gt^2, v = -√(2gh). Time to hit ground from height h is t = √(2h/g). (2) Body projected vertically upwards with initial velocity u: At maximum height, final vertical velocity v = 0. Maximum height attained: H_max = u^2 / (2g). Time of ascent = time of descent: t_up = t_down = u/g. Total time of flight in air: T = 2u/g. The velocity upon returning to the projection level equals -u (same speed, opposite direction).

4. 2D Projectile Motion: Oblique Projection from Ground

A particle launched from ground with speed u at angle θ to the horizontal moves under gravity with zero horizontal acceleration (ax = 0) and constant downward vertical acceleration (ay = -g). Velocity components: initial horizontal ux = u cos θ (remains constant throughout flight); initial vertical uy = u sin θ. Position coordinates at time t: x = (u cos θ)t, y = (u sin θ)t - (1/2)gt^2. Equation of trajectory (parabolic path): y = x tan θ - [g / (2u^2 cos^2 θ)] x^2 = x tan θ [1 - x/R], where R is the horizontal range.

5. Time of Flight, Maximum Height & Horizontal Range

Core projectile parameters: (1) Total Time of Flight: T = (2u sin θ) / g; (2) Maximum Vertical Height: H_max = (u^2 sin^2 θ) / (2g); (3) Horizontal Range: R = (u^2 sin 2θ) / g. Critical properties for entrance exams: Maximum range occurs at launch angle θ = 45°, where R_max = u^2 / g. At this maximum range angle, H_max = R_max / 4. Complementary angles: For two launch angles θ and (90° - θ) at the same projection speed, the horizontal range R is identical, while heights and flight times differ (T1·T2 = 2R/g).

6. Teacher's Exam Hall Hack: Galileo's Law of Odd Numbers

For a body falling freely from rest under uniform gravity, the distances traversed in successive equal intervals of time (1s, 2s, 3s...) bear the ratio of consecutive odd integers: 1 : 3 : 5 : 7 : 9... If a dropped stone travels distance d in the first second, it travels exactly 3d in the second second, and 5d in the third second. Remembering this ratio lets you solve multi-story building drop problems mentally without touching pencil to paper.