Comprehensive Physics Notes – Linear Motion, Forces, Work, Energy & Power

Linear Motion (Motion in 1-D)

  • Definition: motion along a single, fixed direction (straight line)
    • AKA one-dimensional or linear motion
    • Typical journey: start from rest → speed up → cruise (constant or variable speed) → slow down → stop
  • Fundamental quantities
    • Distance
    • Scalar: total length between two points
    • SI unit: meter (m)
    • Displacement
    • Vector: distance covered in a specified direction
    • SI unit: meter (m)
    • Speed
    • Scalar: rate of change of distance
    • \text{Speed}=\frac{\text{Distance}}{\text{Time}}
    • SI unit: \text{m\,s}^{-1}
    • Velocity
    • Vector: rate of change of displacement
    • \text{Velocity}=\frac{\Delta s}{\Delta t}=\frac{\text{Displacement}}{\text{Time}}
    • Units: \text{m\,s}^{-1}
    • Converting km/h to m/s: X\text{ km h}^{-1}=\frac{X\times1000}{3600}\,\text{m s}^{-1}=\frac{5}{18}X\,\text{m s}^{-1}
      • Example: 72\text{ km h}^{-1}=20\text{ m s}^{-1}
    • Acceleration
    • Vector: rate of change of velocity
    • a=\frac{\Delta v}{\Delta t}
    • Units: \text{m\,s}^{-2}
    • Retardation (deceleration): negative acceleration when speed decreases

Graphical Analysis

  • Displacement–Time (s–t) graph
    • Slope =\frac{\Delta s}{\Delta t}=v (velocity)
    • Constant slope → uniform velocity
  • Distance–Time graph
    • Slope gives speed
  • Velocity–Time (v–t) graph
    • Slope =a (acceleration)
    • Patterns
    • OA: constant positive acceleration
    • AB: zero acceleration (constant velocity)
    • BC: constant negative acceleration (retardation)
    • Area under v–t curve =\text{displacement}

Equations of Motion (Uniform Acceleration)

For constant acceleration a, initial velocity u, final velocity v, time t, displacement s:

  1. v=u+at
  2. s=ut+\tfrac12at^2
  3. v^2=u^2+2as
  4. Average velocity method: s=\tfrac12(u+v)t

Worked Example (Horizontal)

Delivery van problem

  • Given: u=0, a1=1.2\,\text{m s}^{-2} for t1=14\,\text{s}
    • v1=u+a1t_1=16.8\,\text{m s}^{-1}
    • s1=\tfrac12 a1t_1^2=117.6\,\text{m}
  • Cruise: t_2=50\,\text{s} at v=16.8\,\text{m s}^{-1}
    • s2=v t2=840\,\text{m}
  • Deceleration: a_3=-2.0\,\text{m s}^{-2} to rest
    • t3=\frac{v}{|a3|}=8.4\,\text{s}
    • s3=\tfrac12 v t3=70.6\,\text{m}
  • Total distance s{tot}=s1+s2+s3\approx1029\,\text{m}

Vertical Linear Motion (Free Fall & Projection)

  • Acceleration due to gravity g\approx10\,\text{m s}^{-2}
    • Positive when object moves downward; negative when upward
  • Replace a by \pm g in motion equations
    • Falling: v=u+gt, s=ut+\tfrac12gt^2, v^2=u^2+2gs
    • Upward projection: v=u-gt, s=ut-\tfrac12gt^2, v^2=u^2-2gs
  • Example: Ball projected upward with u=30\,\text{m s}^{-1}
    • Max height: v=0\Rightarrow0=u^2-2gs\implies s=45\,\text{m}
    • Time to top: t=\frac{u}{g}=3\,\text{s} ⇒ round-trip time =6\,\text{s}

Force

  • Definition: push or pull that causes (or attempts) motion/change of state
  • Vector; SI unit: Newton (N)
  • Common types discussed
    • Tension, Centripetal, Centrifugal, Friction, Magnetic, Electrostatic, Up-thrust, Surface tension, Gravitational (weight), Adhesion, Cohesion
  • Components of a force example
    • Lawn-mower: F=40\,\text N at 50^{\circ} below horizontal
    • F_H=F\cos50^{\circ}\approx25.7\,\text N
    • F_V=F\sin50^{\circ}\approx30.6\,\text N

Resultant Force

  • Concurrent forces → single equivalent (resultant) force
    • Example: 5\,\text N right + 12\,\text N down ⇒ magnitude 13\,\text N, direction 292.6^{\circ} (from +x axis)

Newton’s Laws of Motion

  1. Law of Inertia
    • Body remains at rest/uniform straight-line motion unless acted on by external unbalanced force
    • Mass measures inertia; greater mass → harder to change motion
    • Bus example: passengers lurch forward when bus stops abruptly
  2. Law of Acceleration
    • Rate of change of momentum ∝ resultant force; direction same as force
    • Derivation: F=ma, Impulse =Ft=m(v-u) (unit: N·s)
    • Example: 72\,\text{km h}^{-1}\,(20\,\text{m s}^{-1}) car, m=1200\,\text{kg} stopped in 4\,\text{s}
      • a=-5\,\text{m s}^{-2}, F=-6000\,\text N, s=40\,\text m
  3. Law of Action–Reaction
    • For every action there is equal and opposite reaction
    • Book on table: weight mg downward, normal reaction equal upward

Weight (Gravitational Force)

  • W=mg
  • Varies with g, whereas mass is invariant
  • Example: 60-kg man → W=600\,\text N if g=10\,\text{m s}^{-2}

Friction

  • Opposes relative motion between surfaces
  • Types: Static, Limiting, Dynamic (Kinetic)
    • F{static}>F{limiting}>F_{kinetic}
  • Laws (solids)
    1. Opposes motion
    2. F\propto R (normal reaction) ⇒ F=\mu R=\mu mg
    • \mu (coefficient of friction): 0<\mu<1; indicates roughness
    1. Independent of contact area (if R constant)
    2. Kinetic friction ≈ independent of relative velocity
  • Dependence
    • Increase \mu by roughening surfaces (e.g., shoe treads)
    • Increase R by adding load
  • Sample calc (slide 28)
    • 10-kg block, \mu=0.4, applied 200\,\text N horizontal
    • Ff=\mu mg=40\,\text N, F{res}=160\,\text N, a=\frac{F}{m}=16\,\text{m s}^{-2}

Work

  • Conditions: force applied, displacement occurs, force has component along displacement
  • Scalar; unit: Joule (J)
  • General formula
    • W=|\vec F|\,|\vec d|\cos\theta (\theta between \vec F & displacement)
    • Special: W=Fd when force parallel to motion
  • Example list
    • 20-N force accelerates block: work =200\,\text J
    • Dragging with 60-N force at 30^{\circ} for 50 m: W=60\times50\cos30^{\circ}=2600\,\text J
    • Lifting 12-kg crate 3 m: W=mgh=12\times10\times3=360\,\text J (≈353 J given rounding g)
    • Wagon stopped by opposite 60-N force in 2 s: work done against motion =900\,\text J

Energy

  • Forms: Electrical, Sound, Solar, Heat, Wind, Geothermal, Tidal, Nuclear, Chemical, Mechanical (Kinetic & Potential) … all inter-convertible
  • Conservation Principle: energy cannot be created/destroyed, only transformed (e.g., search-light: chemical → electrical → light + heat)

Kinetic Energy (KE)

  • KE=\tfrac12 mv^2
  • Derived via work–energy theorem W=\Delta KE
  • Examples
    • 1-kg trolley @ 2 m/s: 2\,\text J
    • 2-g bullet @ 400 m/s: 0.002\times400^2/2=160\,\text J
    • 500-kg car @ 72 km/h (20 m/s): 0.5\times500\times20^2=100,000\,\text J
    • Velocity from KE: v=\sqrt{\frac{2KE}{m}} ⇒ 1-kg object w/200 J ⇒ v=20\,\text{m s}^{-1}

Potential Energy (PE)

  • Gravitational: PE=mgh (h relative to reference level)
  • Example: 50-g object raised 10 m ⇒ PE=5\,\text J
  • Falling 2-kg object from 50 m
    1. Start: PE=1000\,\text J, KE=0
    2. Halfway (25 m): PE=500\,\text J, KE=500\,\text J (using v^2=2gs)
    3. Ground: PE=0, KE=1000\,\text J
    4. PE+KE constant (1000 J) → illustrates energy conservation
  • Additional question: 5-kg body, KE just before ground 1000 J
    • Loss of PE = gain in KE = 1000 J
    • Height fallen h=\frac{PE}{mg}=\frac{1000}{5\times10}=20\,\text m

Power

  • Rate of doing work or transferring energy
    • P=\frac{W}{t}=\frac{Fs}{t}=Fv (if force parallel to velocity)
    • Unit: Watt (W) = J/s
  • Stair-climb example
    • Boy: m=40\,\text{kg}, 12 steps of 0.15\,\text m each (total h=1.8\,\text m) in 30\,\text s
    • Work against gravity: W=mgh=40\times10\times1.8=720\,\text J
    • Power: P=\frac{720}{30}=24\,\text W

Connections & Implications

  • Equations of motion underpin practical vehicle design, sports, safety engineering
  • Graph interpretation vital for experimental diagnostics (e.g., motion sensors, data loggers)
  • Understanding friction guides material selection, tread design, lubrication ethics (energy efficiency, wear reduction)
  • Conservation of energy foundational for sustainable engineering & physics: informs power generation choices, efficiency calculations, environmental impact assessments.