For projectile on H.P. ⋆ Time of flight $t = {2u sinα}/{g}$ ⋆ Horizontal Range $R = {u^2 sin2α}/{g}$ when α = 45°, Range is maximum: $R_{max} = {u^2}/{g}$ ⋆ Max Height $H_{max} = {u^2 sin^2 α }/{2g}$ ⋆ Equation of path (Trajectory) y = x tanα – ${g x^2}/{2 u^2 cos^2 α}$ ⋆ When projectile is projected with a horizontal velocity from top of a tower of heigh h $y = – {g x^2}/{2 u^2}$ [as α = 0] -h = $-{g x^2}/{2 u^2}$ Horizontal distance, $x = v_o √{{2h}/{g}}$ Time of flight, t = $√{{2h}/{g}}$ ⋆ Same range, ...
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$\table Direction,Velocity-Component,Acceleration-Component;x,v_x={dx}/{dt},a_x={dv_x}/{dt};y,v_y = {dy}/{dt},a_y = {dv_y}/{dt};Tangential,v_T = V, a_T = {dv}/{dt};Normal, v_N = 0, a_N = {v^2}/{ρ};Radial, v_r = r’,a_r = r” – r θ’^2;Transverse, v_θ = r θ’, a_θ = 2r’θ’ + r θ”$ Note: ⋆ $a_T$ represents change in speed while $a_N$ represents change in direction. ⋆ At point of inflection ρ -> ∞ so $a_N$ = 0 ⋆ Always use kinematic equations with $a_T$ in curvilinear motion Methods to find Radius of curvature (1) When speed v and $a_N$ are known $|ρ| = {v^2}/{a_N}$ (2) When path equation y = f(x) is given, $|ρ| = {[a + ...
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No important or extra ordinary formula needed in this chapter
Relative velocity of A w.r.t. B => A observed from B and B is stationary $\ov{V_{A|B}} =\ov{V_{A}} – \ov{V_{B}}$ ∙Two particles, velocities parallel Same direction: $|V_{A|B}| = v_A – v_B$ in magnitude Opposite direction: $|V_{A|B}| = v_A + v_B$ in magnitude ∙ Two velocities perpendicular $|V_{A|B}| = √{{V^2_A} + {V^2_B}}$ $tanθ = {V_B}/{V_A}$
Displacement (s) = $x_t – x_o$ $d↙{(0 -> t)} = |x_y – x_o|$ Instantaneous velocity = dx/dt Displacement ≦ Distance Velocity ≦ speed a = dv/dt OR a = a ${dv}/{ds}$ Jerk: Rate of change of acceleration w.r.t time $J = {da}/{dt}$ Types of motion and kinematic equations (1) Uniform motion: v is constant and a = 0 s= v x t (2)Uniformly accelerated motion: a os constant and J = 0 V = u + at, s = ut + $1/2$ a $t^2$ $v^2 = u^2 + 2as$ (3)Motion under gravity: g is constant v = u – gt ...
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Magnitude of Resultant: $R = √{P^2 + Q^2 + 2PQ cosθ}$ Direction of Resultant: $tan α = {Q sinθ}/{P + Q cos θ}$ Resultant of two or more forces: $R = √{(ΣF_x)^2 + (ΣF_y)^2} $ Direction of two or more forces: $tan θ = {ΣF_y}/{ΣF_x}$ $ΣF_x$ = Algebric sum of all x components (or x component of resultant) $ΣF_y$ = Algebric sum of all y components (or y component of resultant) θ = Angle of ‘R’ with x-axis When an object is inclined on a plane at angle θ Component along plane -> mg sinθ Component perpendicular to plane -> ...
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