MOTION IN A STRAIGHT LINE — LECTURE 02 (NKC Sir)

Recap of Lecture 01

  • Vector subtraction reminder: \overrightarrow{OA}-\overrightarrow{OB}=\overrightarrow{BA}
  • Displacement (\Delta x), instantaneous velocity v and average velocity v_{avg} relations
    • v = \dfrac{dx}{dt}
    • v_{avg}=\dfrac{\Delta x}{\Delta t}
  • Instantaneous speed = magnitude of v ; always positive.

Differential Relations for 1-D Motion

  • Acceleration (instantaneous)
    • a = \dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}
    • Chain–rule form (very useful for variable-velocity questions): a = v\,\dfrac{dv}{dx}
  • Hierarchy of kinematical quantities (along a straight line)
    • x(t) \xrightarrow{\;d/dt\;} v(t) \xrightarrow{\;d/dt\;} a(t)
    • Going back: v(t)=\displaystyle\int a\,dt,\;x(t)=\displaystyle\int v\,dt

Speed vs. Velocity

  • Speed = rate of change of distance; scalar; SI unit \text{m\,s}^{-1}.
  • Velocity = rate of change of displacement; vector; can be +,-,0 depending on chosen axis.
  • Sign convention (commonly used):
    • + upward/rightward, - downward/leftward.

Varieties of Speed

  • Uniform speed : equal distances in equal times.
  • Non-uniform speed : unequal distances in equal times.
  • Average speed = \dfrac{\text{total distance}}{\text{total time}}.
  • Instantaneous speed read by a speedometer =|v| at that instant.

Worked-out Concept Questions

1. Four persons at the corners of a square (dog-chase problem)

  • Side of square d, everyone runs with constant speed u, each always faces the next person.
  • Path: logarithmic spiral into the centre.
  • Relative radial speed toward centre = u.
  • Time to meet t = \dfrac{d}{u} (Option A).
    • Useful trick: radial component only matters.

2. Point on a rolling wheel (half revolution)

  • Radius R, initial point was touching the ground.
  • After half rotation, centre translates by \pi R, point is diametrically opposite (height 2R above initial level).
  • Resultant displacement magnitude
    \displaystyle \Delta r = \sqrt{(\pi R)^2 + (2R)^2}= R\sqrt{\pi^2+4} (Option A).

3. Athlete running around a circular track in sequence A→B→A→B

  • Track radius r.
  • One complete diameter AB traversed thrice ⇒ total distance 3\pi r.
  • Net displacement after ABAB: from A back to B = diameter 2r.
  • Correct pair \bigl(3\pi r,\;2r\bigr) (None of the first four listed options matched, hence answer marked "None of these").

4. Integration example (JEE Main 2021)

  • Given v=v_0+gt+Ft^2, x(0)=0.
  • x=\displaystyle\int0^1 v\,dt = v0 t + \dfrac{g t^2}{2}+\dfrac{F t^3}{3}\Big|{0}^{1}=v0+\dfrac{g}{2}+\dfrac{F}{3}.

5. 3-D displacement (6 m N, 8 m E, 10 m up)

  • Magnitude =\sqrt{6^2+8^2+10^2}=10\sqrt2\,\text{m} (Option A).

6. Average speed on piece-wise segments (AB, BC, CD)

  • Equal lengths AB=BC=x, AD=3x \Rightarrow CD = x.
  • Time on each: ti = \dfrac{x}{vi}.
  • v{avg}=\dfrac{4x}{\frac{x}{v1}+\frac{x}{v2}+\frac{x}{v3}}=\dfrac{4v1v2v3}{v1v2+v2v3+v3v_1}.

7. Half-distance with different speeds

  • First half (x/2) at 6 m s$^{-1}$
  • Remaining half in two equal time slots at 3 m s$^{-1}$ and 15 m s$^{-1}$.
  • Average speed derived to be 7.2\,\text{m\,s}^{-1} (Option C).

8. Policeman–speeder chase

  • Data: reaction delay 2\,\text{s}, policeman acceleration phase 12\,\text{s} to 150\,\text{km h}^{-1}, meeting point 1.5\,\text{km} ahead.
  • Computed speeder’s constant speed \approx 122.7\,\text{km h}^{-1}.

9. When is the particle retarding?

  • v=t^2-t \;\Rightarrow\; a=\dfrac{dv}{dt}=2t-1.
  • Retardation (a < 0) for t<\dfrac12\,\text{s}.

Vector (Scalar-form) Equations of Motion (uniform a, straight line)

  • v = u + a t
  • s = ut + \dfrac12 a t^2
  • v^2 = u^2 + 2 a s
  • Valid only for: straight-line motion with constant acceleration.

Displacement in the n-th Second

  • General: S_n = S(t=n) - S(t=n-1)
  • Using s=ut+\dfrac12 a t^2 ⇒
    \boxed{S_n = u + \dfrac{a}{2}\,(2n-1)}
Special Series Results (start from rest, u=0)
  • Distance in successive full seconds: 1^2 : 2^2 : 3^2 :\dots =1:4:9:\dots
  • Distance in 1st, 2nd, 3rd… individual seconds: 1:3:5:\dots (arithmetic progression).
  • Velocities after 1, 2, 3 s …: 1:2:3:\dots.

Application Example: Displacement in (p^2-p+1)-th Second

  • Let S1 = displacement in first (p-1) s, S2 = displacement in first p s.
  • S1+S2 = \dfrac{a}{2}\bigl[(p-1)^2 + p^2\bigr] = \dfrac{a}{2}(2p^2-2p+1)
  • S_{\,(p^2-p+1)} = \dfrac{a}{2}(2p^2-2p+1)
  • Therefore, required displacement = S1+S2 (Option A).

Quick Problem-Solving Heuristics

  • Always pick a positive axis and stick to it; let signs carry physical meaning.
  • For chase/spiral problems, project velocity along the radial line to find closing speed.
  • When speed is given as a function of time, integrate once for displacement; when acceleration is needed, differentiate.
  • Piece-wise motion → write individual ti’s, then v{avg}=\dfrac{\Sigma si}{\Sigma ti}.
  • For cyclic or rolling objects, separate translation of centre-of-mass and rotation about centre.

Ethical / Exam-oriented Notes

  • Dimensional sanity checks prevent algebraic mistakes.
  • Clearly state assumptions (e.g.
    “uniform a” or “speed constant, direction changing”).
  • JEE often hides constant-speed yet changing-direction cases (speed ≠ velocity magnitude!).
  • Keep calculators in angle–rad (for \pi) while working symbolic forms; final answers normally as multiples of \pi or radicals.

Homework recommended by NKC Sir

  • Solve DW-module Prarambh Q-1 → 16.
  • HCV Kinematics Ex-1 → 20.
  • Attempt DPP & class-sheet questions.