Motion and Energy

Newton's Laws of Motion

Momentum

Product of an object's mass and velocity; formula p = mv; units are kg·m/s or N·s; a vector quantity

Impulse

Effect of a force acting over a time interval; formula J = F_net,avg × Δt; units are N·s or kg·m/s; a vector quantity

Impulse-Momentum Theorem

The impulse acting on an object equals its change in momentum; J = Δp or F_net,avg × Δt = Δp

Change in Momentum

Requires a net force to occur; calculated as Δp = F_net,avg × Δt; depends on both force magnitude and time duration

Stopping Time and Cushioning

Increasing the time interval reduces the required average force; principle used in airbags, bent knees when jumping, and protective padding; |F_net,avg| = |Δp| / Δt

Conservation of Momentum

In collisions where external force is negligible, total momentum before equals total momentum after; p_before = p_after; applies to both elastic and inelastic collisions

Kinetic Energy

Energy an object possesses because of its motion; formula KE = ½mv²; units are Joules (J); a scalar quantity; depends on mass and square of velocity

Gravitational Potential Energy

Energy an object possesses because of its position above Earth's surface; formula PE = mgh; units are Joules (J); a scalar quantity; relative to a reference level

Mechanical Energy

Sum of an object's kinetic energy and gravitational potential energy; formula ME = KE + PE or ME = ½mv² + mgh; units are Joules (J)

Conservation of Energy

In absence of friction and air resistance, total mechanical energy remains constant; ME_initial = ME_final; energy transforms between kinetic and potential forms but total stays the same

Energy Transformation

Energy changes form as objects move; falling object converts PE to KE; rising object converts KE to PE; pendulum continuously transforms KE and PE; friction converts mechanical energy to thermal energy and sound

Momentum vs Kinetic Energy: Vector Property

Momentum is a vector quantity (has direction); kinetic energy is a scalar quantity (no direction)

Momentum vs Kinetic Energy: Units

Momentum units are kg·m/s or N·s; kinetic energy units are Joules (J) or kg·m²/s²

Momentum vs Kinetic Energy: Proportionality

Momentum is proportional to velocity (p ∝ v); kinetic energy is proportional to velocity squared (KE ∝ v²)

Momentum vs Kinetic Energy: Conservation

Momentum is conserved in all collisions when external forces are negligible; kinetic energy is only conserved in elastic collisions, not in inelastic collisions

Momentum vs Kinetic Energy: Velocity Change

If velocity doubles, momentum doubles; if velocity doubles, kinetic energy quadruples (factor of 4)

Momentum vs Kinetic Energy: Same Kinetic Energy

Two objects with the same kinetic energy can have different momenta; light fast object and heavy slow object can have equal KE but different p

Momentum vs Kinetic Energy: Collision Application

Momentum always conserved in collisions; kinetic energy may or may not be conserved depending on collision type; lost kinetic energy converts to heat, sound, and deformation

Elastic Collision

Collision where both momentum and kinetic energy are conserved; objects may bounce off each other

Inelastic Collision

Collision where momentum is conserved but kinetic energy is not; kinetic energy converts to heat, sound, and deformation; objects may stick together

Perfectly Inelastic Collision

Objects stick together after collision; most extreme form of inelastic collision; momentum conserved but significant kinetic energy loss

Double Velocity Effect on Momentum

If velocity doubles, momentum doubles; directly proportional relationship

Double Velocity Effect on Kinetic Energy

If velocity doubles, kinetic energy increases by factor of 4; KE is proportional to v² so (2v)² = 4v²

Airbags in Cars

Safety device that increases stopping time during collision; longer Δt reduces the average force on passengers; applies principle |F_net,avg| = |Δp| / Δt

Cushioning Principle

Increasing contact time with ground or surface reduces impact force; bent knees when jumping, padding in helmets, and soft surfaces all increase Δt to reduce force

Average Force Calculation

Formula: |F_net,avg| = |Δp| / Δt; if stopping distance same, longer stopping time results in smaller force

Gravitational Potential Energy Reference Level

PE is always relative to a chosen reference point (usually ground level = 0); same object has different PE values depending on reference level chosen

Double Mass Effect on Potential Energy

If mass doubles at same height, potential energy doubles; directly proportional to mass since PE = mgh

Double Height Effect on Potential Energy

If height doubles at same mass, potential energy doubles; directly proportional to height since PE = mgh

Double Mass Effect on Kinetic Energy

If mass doubles at same velocity, kinetic energy doubles; directly proportional to mass since KE = ½mv²

Double Mass Effect on Momentum

If mass doubles at same velocity, momentum doubles; directly proportional to mass since p = mv

Pendulum at Lowest Point

Has maximum kinetic energy and zero potential energy (if taking lowest point as reference); moving fastest

Pendulum at Highest Point

Has maximum potential energy and zero kinetic energy; momentarily at rest before reversing direction

Pendulum Energy Cycle

Continuously transforms between KE at bottom and PE at top; in ideal conditions total mechanical energy stays constant throughout motion

Falling Object Energy Pattern

At top: maximum PE, zero KE; during fall: PE decreases and KE increases; at bottom: zero PE, maximum KE; total ME constant if no friction

Rising Object Energy Pattern

At launch: maximum KE, zero PE; during rise: KE decreases and PE increases; at peak: zero KE, maximum PE; at launch point KE equals PE at peak

Rollercoaster at Top

Has maximum potential energy; if starting from rest, all mechanical energy is potential energy

Rollercoaster at Bottom

Has maximum kinetic energy; if reference level at bottom, all mechanical energy is kinetic energy

Energy Dissipation by Friction

Friction and air resistance convert mechanical energy to thermal energy (heat) and sound energy; mechanical energy decreases over time; object eventually stops

Real-World vs Ideal Situations

Ideal: no friction or air resistance, mechanical energy conserved; real world: friction present, mechanical energy dissipated as heat and sound

Mechanical Energy with Friction

Formula: ME_f = ME_i - E_dissipated; final mechanical energy equals initial minus energy lost to friction and air resistance

Stopped Object Kinetic Energy

Once at rest, kinetic energy is zero; all motion energy has been converted to other forms (heat, sound, deformation)

Velocity from Energy Conservation

Can calculate final velocity using ½mv_i² + mgh_i = ½mv_f² + mgh_f when friction is negligible

Height from Energy Conservation

Can calculate maximum height using energy conservation; at peak height velocity is zero so all kinetic energy converts to potential energy

Terminal Velocity

Maximum velocity reached when air resistance force equals gravitational force; object no longer accelerates; kinetic energy no longer increases

Momentum Calculation Example

1500 kg car at 20 m/s: p = (1500)(20) = 30,000 kg·m/s

Kinetic Energy Calculation Example

2 kg ball at 10 m/s: KE = ½(2)(10)² = 100 J

Potential Energy Calculation Example

5 kg box 2 meters above ground: PE = (5)(9.8)(2) = 98 J

Mechanical Energy Calculation Example

2 kg ball moving 5 m/s at 3 meters height: KE = 25 J, PE = 58.8 J, ME = 83.8 J

Collision Momentum Conservation Example

1000 kg car at 10 m/s hits stationary 1000 kg car; they stick together; final velocity is 5 m/s (momentum conserved: 10,000 = 2000v_f)

Falling Stone Velocity Example

2 kg stone dropped from 45 meters reaches velocity of 29.7 m/s at ground (using ½mv² = mgh, v = √(2gh))

Energy Transformation Pattern Falling

PE at top converts to KE during fall; PE_initial = KE_final if no friction

Energy Transformation Pattern Rising

KE at launch converts to PE during rise; KE_initial = PE_final at peak height

Pendulum Velocity Calculation

1.5 kg bob pulled horizontal (1 meter high) swings to bottom with velocity 4.43 m/s (using mgh = ½mv²)

Roller Coaster Velocity No Friction

500 kg car from 50 meter height reaches 31.3 m/s at bottom (using ½mv² = mgh)

Roller Coaster Velocity With Friction

Same car with 50,000 J dissipated reaches 27.9 m/s instead of 31.3 m/s (reduced due to energy loss)

Stopping Force Baseball Example

0.15 kg baseball at 40 m/s caught in 0.05 seconds requires average force of 120 N (F = Δp / Δt)

Stopping Force Person Example

70 kg person hitting hard concrete (Δt = 0.01 s) experiences 44,100 N force; with cushioning (Δt = 0.5 s) experiences 882 N

Impulse Calculation Example

50 N force acting for 3 seconds produces impulse of 150 N·s (J = F × Δt)

Two Ice Skaters Momentum Example

60 kg person at 2 m/s right, 80 kg person at 1.5 m/s left initially at rest; momentum before equals momentum after (both zero)

Same Velocity Different Mass Kinetic Energy

1000 kg car and 100 kg motorcycle both at 20 m/s; car has 10 times more kinetic energy (KE depends on mass)

Same Kinetic Energy Different Momentum

1 kg at 10 m/s has KE = 50 J and p = 10 kg·m/s; 10 kg at 3.16 m/s has KE = 50 J and p = 31.6 kg·m/s

Velocity Doubling Momentum

Doubling velocity doubles momentum (direct proportion: p = mv)

Velocity Doubling Kinetic Energy

Doubling velocity quadruples kinetic energy (squared relationship: KE = ½m(2v)² = 4(½mv²))

Maximum Potential Energy Point

Highest point of trajectory for thrown object; point furthest from reference level

Maximum Kinetic Energy Point

Lowest point of trajectory or fastest point in motion; greatest speed

Energy at Maximum Height

At maximum height velocity is zero, so kinetic energy is zero and all mechanical energy is potential

Equal Kinetic and Potential Energy

Occurs at height h = v_i²/(4g) or halfway to maximum height (where KE = PE)

Conservation of Momentum Condition

System must be isolated; negligible external forces; friction and air resistance typically negligible during collision

Inelastic Collision Energy Loss

Kinetic energy converted to thermal energy, sound energy, deformation energy, and sometimes light energy; mechanical energy appears lost

Elastic vs Inelastic Collision

Elastic: KE conserved, objects bounce; inelastic: KE not conserved, objects may stick or partially deform

Head-On Collision Momentum Example

2000 kg truck at 15 m/s hits 1000 kg car at 20 m/s opposite direction; they lock together at 3.33 m/s in truck's original direction

Work-Energy Relationship

Work done on object changes its kinetic energy; W = ΔKE; larger work produces larger change in kinetic energy

Force and Momentum Change

To change momentum, net force must act; larger forces produce faster momentum changes; F = Δp / Δt

Friction Converting Mechanical Energy

Sliding friction converts kinetic energy to thermal energy; object slows down as mechanical energy is dissipated

Energy Input to System

Adding energy increases mechanical energy; removal of energy decreases mechanical energy

Pendulum Ideal vs Real

Ideal pendulum: oscillates forever with constant mechanical energy; real pendulum: gradually stops as friction dissipates energy

Maximum Height from Initial Velocity

Using ½mv² = mgh, height = v² / (2g); doubling initial velocity quadruples maximum height

Final Velocity from Height

Using ½mv² = mgh, velocity = √(2gh); velocity depends only on height, not on mass

Common Misconception Momentum KE

Momentum ∝ v but KE ∝ v²; these are different relationships, not the same

Reference Level for Potential Energy

Must specify what height equals zero; changing reference level changes all PE values but not ME changes

Collision Momentum Always Conserved

Momentum conserved in all collisions if external forces negligible; true for elastic AND inelastic collisions

Friction Force Direction

Friction force opposes motion; always acts opposite to velocity direction

Energy Dissipation Rate

Energy dissipation depends on friction coefficient and normal force; greater friction dissipates energy faster

Work Done by Friction

Work by friction equals negative kinetic energy change; W_friction = -ΔKE (negative because it opposes motion)

Stopping Distance Relationship

Longer stopping distance allows smaller force; inverse relationship: larger Δt means smaller F for same Δp

Object at Rest Momentum and Energy

Momentum = 0 and kinetic energy = 0; potential energy depends on height above reference level

Elastic Collision Equations

Momentum conserved: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f; kinetic energy conserved: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f²

Perfectly Inelastic Collision Equations

Momentum conserved: m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)vf; objects stick together with same final velocity

Impact Safety Padding

Padding increases contact time during impact; longer Δt reduces average force and prevents injury; same principle as airbags

Sports Applications Momentum

Follow-through in throwing increases impulse time; catching with moving hands increases stopping time; martial arts breaking involves large force over short time

Hydroelectric Power Potential Energy

Water at height has gravitational potential energy; flowing down converts PE to KE; KE drives turbine to generate electricity

Braking System Energy

Kinetic energy of moving vehicle converted to thermal energy (heat) in brake pads and rotors; friction force opposes motion

Wind Turbine Kinetic Energy

Moving air has kinetic energy; turbine blades convert KE of wind into rotational motion to generate electricity

Solar Panel Energy

Light energy converted to electrical energy; different energy form conversion

Rolling Motion Energy

Kinetic energy in rolling object includes both translational and rotational motion; not covered in basic KE = ½mv²

Collision in Two Dimensions

Momentum conserved in each direction separately (x and y components); overall momentum vector conserved

System vs Object

System: collection of objects; momentum conserved for system if no external forces; momentum may transfer between objects

Center of Mass Momentum

Total momentum of system equals mass times velocity of center of mass; p_total = M_total × v_center_of_mass

Recoil Momentum

When object propels something, recoil occurs in opposite direction; total momentum before launch equals zero; momentum conserved