Kinematics: Rectilinear & Projectile Motion

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It forms the backbone of JEE Physics, connecting directly to Newton's Laws, Work-Energy, and Rotational Motion.

Rectilinear Motion deals with motion along a straight line. The three fundamental kinematic equations (valid only for constant acceleration) are:

1. v = u + at

   • v: final velocity (m/s), u: initial velocity (m/s), a: acceleration (m/$s^{2}$), t: time (s)
   • Dimensional formula: [LT^{-1}] = [LT^{-1}] + [LT^{-2}][T]

2. s = ut + ($\frac{1}{2}$)$at^{2}$

   • s: displacement (m), [L] = [LT^{-1}][T] + [LT^{-2}][$T^{2}$]

3. $v^{2}$ = $u^{2}$ + 2as

   • [$L^{2}$T^{-2}] = [$L^{2}$T^{-2}] + [LT^{-2}][L]

Displacement in nth second: $s_{n}$ = u + a(2n - 1)/2. This is NOT distance; it can be negative.

Projectile Motion is two-dimensional motion under constant gravitational acceleration g = 9.8 m/$s^{2}$ (downward). A projectile launched at angle $\theta$ with speed u has:

• Horizontal component: $u_{x}$ = u cos($\theta$), $a_{x}$ = 0
• Vertical component: $u_{y}$ = u sin($\theta$), $a_{y}$ = -g (taking upward as positive)

Key Projectile Formulae:

• Time of flight: T = 2u sin($\theta$)/g [T] = [LT^{-1}]/[LT^{-2}] = [T]
• Maximum height: H = $u^{2} \; sin^{2}$($\theta$)/(2g) [L] = [$L^{2}$T^{-2}]/[LT^{-2}] = [L]
• Range: R = $u^{2}$ sin(2*$\theta$)/g [L]
• Maximum range at $\theta$ = 45 degrees: $R_{max}$ = $u^{2}$/g

Relative Motion: v_{A/B} = $v_{A}$ - $v_{B}$. For rain-man problems, draw the velocity triangle.

Graphs: The slope of x-t graph gives velocity; slope of v-t graph gives acceleration; area under v-t graph gives displacement; area under a-t graph gives change in velocity.

The key problem-solving concept is: always define a sign convention FIRST (typically upward/rightward = positive), resolve vectors into components, and track signs meticulously through every step.

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