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: $u=0$, $a<em>1=1.2\,\text{m s}^{-2}$ for $t</em>1=14\,\text{s}$
  • $v<em>1=u+a</em>1t_1=16.8\,\text{m s}^{-1}$
  • $s<em>1=\tfrac12 a</em>1t_1^2=117.6\,\text{m}$
• Cruise: $t_2=50\,\text{s}$ at $v=16.8\,\text{m s}^{-1}$
  • $s<em>2=v t</em>2=840\,\text{m}$
• Deceleration: $a_3=-2.0\,\text{m s}^{-2}$ to rest
  • $t<em>3=\frac{v}{|a</em>3|}=8.4\,\text{s}$
  • $s<em>3=\tfrac12 v t</em>3=70.6\,\text{m}$
• Total distance $s<em>{tot}=s</em>1+s<em>2+s</em>3\approx1029\,\text{m}$

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
  • $F<em>f=\mu mg=40\,\text N$, $F</em>{res}=160\,\text N$, $a=\frac{F}{m}=16\,\text{m s}^{-2}$

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.