Points:

• Amount,length,time,luminous intensity, mass, temperature, current are the fundamental physical quantities.
• Same units in every term on either side of =
• When round determining digit is 5 and the truncated digits are 0 then the number is rounded up if the number to remain is odd and truncated if the number to remain is even.
• For ± least decimal places, for */ least significant figure
• If something else pushes or pulls an object, its velocity will change

Trivia:

• Df(x) = f'(x)
• You can move vectors
• Motion of object with respect to another object is relative motion
• When relative velocity is towards the reference point it is called velocity of approach
• Forces are either contact or non-contact

Non-formulae:

• a = a_m ± Δa {Δa = Σ|a_m-a_i|}
• nVSD = (n-1)MSD
• LC = VC = 1MSD-1VSD = (1/n)MSD
• LC = pitch/number of divisions
• |A+B| = sqrt(|A|² + |B|² + 2|A||B|cosθ)
• Angle of A+B with A = tan^-1(Bsinθ/(A + Bcosθ))
• A•B = |A||B|cosθ = A_xB_x + A_yB_y + A_zB_z
• AxB = det(i j k , A_x A_y A_z, B_x B_y B_z) = |A||B|sinθ * direction
• v = u + at
• s = ut + at²/2
• v²- u² = 2as
• s_n = u + a(2n-1)/2
• H = (usinθ)²/2g = u_y²/2g
• R = u²sin2θ/g = 4Hcotθ
• R (for horizontal projectile) = u_xu_y/g = u_xsqrt(H/2g)
• Minimum speed to hit (x,y) = u_min = sqrt(g(y+sqrt(x² + y²)))
• Minimum angle = tan^-1[u_min²/gx]

Conditional:

Elastic collision with wall

If a wall is in the path of a projectile at a distance x from the starting point and the collision is elastic then:

1. t_f and H remain the same , distance covered after the collision will be in opposite x direction and same y direction as distance covered before collision
2. If x ≥ R/2 landing spot is at distance 2x-R from the launch point in the same x direction as initial velocity
3. If x ≤ R/2 landing spot is at distance R-2x from the launch point in the opposite x direction to initial velocity

Projectile on inclined plane

For up the plane:

• The angle of inclination is α
• The acute angle of projectile be θ
• t_f = 2usinθ/gcosα
• R = u²[sin(2θ+α)-sinα]/gcos²α
• R is max when θ = (π/4) - (α/2) and, R_max = u²/g(1+sinα)
• Max distance from inclined plane = u²sin²(θ)/2gcosα

For down the plane replace α with -α to get the corrected expressions

Man swimming in river

• V_man = V_man|river + V_river {v = |V_man|river|}
• Time taken to cross river, t = d/vsinθ , θ = angle between V_man|river and bank of the river
• Drift = (vcosθ + u)*t (u=|V_river|)
• For drift to be zero , cosθ = -u/v , t = d/sqrt(v² - u²)
• If v<u minimum non-zero drift is when cosθ = -v/u
• For swimming in a specific direction,
  Angle of V_man|river with direction of swim be α
  vsinα = usinθ , θ is angle of direction of swim with river bank

Minimum distance between 2 moving bodies:

Initial distance = L
Angle made between V_rel and initial displacement = θ
Angles made by the moving bodies be α & β and velocities be V_a , V_b

• Minimum distance = Lsinθ
• |V_rel| = sqrt(|V_a|^2 + |V_b|^2 +2|V_a||V_b|cos(α + β))
• Time to reach minimum separation is Lcosθ/|V_rel|

For collision:

• |V_a|sinα = |V_b|sinβ
• Time taken to collide = L/(|V_a|α + |V_b|cosβ)

Formulae:

• Zero error = -zero correction
• Length = reading + zero correction
• (1+x)ⁿ = 1 + nx + n(n-1)x/2! + n(n-1)(n-2)x/3! + ... , |x|<1
• a/sin(A) = b/sin(B) = c/sin(C)
• A = |A|*dir(A)
• |A| = sqrt(A_x² + A_y² + A_z²)
• D_A|B = D_A - D_B
• t_f = 2(usinθ)/g
• t_d = sqrt(H/2g)
• For man in rain to remain dry, he should hold his umbrella at an angle of θ = tan^-1(V_m/V_r)
• a = F/m

Tips:

• Use dot product to find angle between vectors
• Verify if graph given is trajectory or distance-time graph
• For projectile at a height, find t_f using s = ut + at²/2
• For finding shortest distance, consider relative motion
• For collisions, the trajectories should intersect and at the point of intersection the times must be the same

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