Dynamics: Kinetics - Cheat Sheet
Your last-minute revision companion for kinetics. Master Newton's second law, work–energy, impulse–momentum, and impact. Every formula, definition, and pitfall you need in the final 30 minutes.
Newton's Second Law & Force Analysis
Formulas
∑F = ma
Meaning
∑F = net force (N); m = mass (kg); a = acceleration (m/s²)
Watch out
Use MASS (kg), not weight (N). Friction always opposes motion: F_friction = −μN.
When to use
When forces and acceleration are involved; draw FBD first, resolve along direction of motion.
F_friction = μN
Meaning
μ = coefficient of friction (dimensionless); N = normal force (N)
Watch out
Friction acts OPPOSITE to direction of motion. Do not forget to find N first (sum perpendicular forces).
When to use
Always include on horizontal/inclined surfaces with sliding. Static friction ≤ μ_s·N; kinetic friction = μ_k·N.
N = mg cos(θ) [on incline]; N = mg [on horizontal]
Meaning
θ = angle of incline; g = 9.81 m/s²
Watch out
On an incline, the component DOWN the slope is mg·sin(θ), not used in normal force.
When to use
Before calculating friction force on any surface.
F_net_along_incline = mg sin(θ) − μ mg cos(θ)
Meaning
Net force down a rough incline
Watch out
Include friction term; do not forget cos(θ) in friction calculation.
When to use
Block on incline with friction; then a = F_net/m.
Common values
Value
9.81 m/s²
Symbol
g
Quantity
Standard gravity
Value
μ ≈ 0.15–0.25
Symbol
μ
Quantity
Coefficient of friction — steel on steel
Value
μ ≈ 0.6–0.85
Symbol
μ
Quantity
Coefficient of friction — rubber on concrete
Important facts
- Always draw FBD before writing ∑F = ma.
- Resolve forces along and perpendicular to motion direction.
- Friction force is kinetic (constant) once sliding begins; static friction is self-adjusting up to μ_s·N.
- On horizontal surface: N = mg exactly; on incline: N = mg cos(θ).
- Tension in a rope or cable acts along the rope; assume massless rope unless stated.
Key definitions
D'Alembert's Principle
Example
A truck accelerating at 2 m/s² — add reverse inertia force to treat as statics.
Inertia force −ma added to FBD treats dynamics problem as pseudo-equilibrium: ∑F − ma = 0.
Free Body Diagram (FBD)
Example
Block on floor: weight down, normal up, friction left, push right.
Isolated sketch of body showing ALL external forces (weight, normal, friction, applied force).
Mass vs. Weight
Example
A 50 kg object weighs 50 × 9.81 = 490.5 N on Earth.
Mass m (kg) is amount of matter; weight W = mg (N) is gravitational force on mass.
Diagrams to know
- Free body diagram (FBD) with all forces labeled
- Inclined plane with force components: mg sin(θ) parallel, mg cos(θ) perpendicular
- Friction force vector opposing motion direction
Work–Energy Principle
Formulas
U₁→₂ = ½m(v₂²) − ½m(v₁²) = ΔKE
Meaning
U = net work done (J); v₁, v₂ = initial and final speeds (m/s); m = mass (kg)
Watch out
Use NET work (all forces combined). If multiple forces act, sum their works: U_net = U_F + U_friction + U_gravity.
When to use
Unknown is speed over a distance; time NOT involved. Avoids integration of varying forces.
U = F·d·cos(θ)
Meaning
F = force magnitude (N); d = displacement (m); θ = angle between F and displacement
Watch out
θ = 0° (force along motion) → U positive; θ = 90° (force perpendicular) → U = 0; θ = 180° (force opposes) → U negative.
When to use
Work of a constant force along a path. Component of force along displacement is F·cos(θ).
U_gravity = ±Wh = ±mgh
Meaning
W = weight (N); h = vertical height change (m); + if descending, − if ascending
Watch out
Positive going DOWN. If object rises, U_gravity is NEGATIVE (work done AGAINST gravity).
When to use
Any motion with height change. Conservative force; work depends only on initial and final heights.
U_spring = ½k(x₁² − x₂²)
Meaning
k = spring stiffness (N/m); x₁, x₂ = compression/extension from natural length (m)
Watch out
This is −ΔPE_spring. Stored energy is PE = ½kx². Do NOT use ½kx² as work; use ½k(x₁² − x₂²).
When to use
Spring returns to natural length (or changes compression); U is the ENERGY RELEASED by spring.
U_friction = −μN·d
Meaning
μ = coefficient of kinetic friction; N = normal force (N); d = distance of sliding (m)
Watch out
ALWAYS NEGATIVE. Common error: treating friction work as positive.
When to use
Always negative; friction opposes motion. d is the actual distance traveled.
Common values
Value
1 N·m = 1 kg·m²/s²
Symbol
J
Quantity
1 Joule
Important facts
- Work–energy is fastest when the unknown is a speed and distance is known.
- Friction always does negative work: −μNd (not −μN·d²).
- Gravitational potential energy: PE_gravity = mgh (measured from reference level).
- Spring potential energy: PE_spring = ½kx² (x from natural length).
- Normal force does zero work on a horizontal surface (perpendicular to motion).
- Energy is conserved with no friction/non-conservative forces: total KE + PE = constant.
Key definitions
Kinetic Energy (KE)
Example
A 10 kg object at 5 m/s has KE = ½(10)(25) = 125 J.
Energy of motion: KE = ½mv²; always non-negative.
Work (U)
Example
Pushing a box 5 m with 20 N force: U = 20(5) = 100 J.
Energy transferred by a force: U = ∫F·ds; positive if force aids motion, negative if opposes.
Conservative Force
Example
Height change from point A to B gives same gravitational work regardless of path taken.
Force whose work depends only on initial and final positions, not the path (gravity, spring).
Mechanical Energy Conservation
Example
Pendulum: at bottom KE max, PE min; at top KE min, PE max; total always same.
If only conservative forces act: KE₁ + PE₁ = KE₂ + PE₂ (total energy constant).
Diagrams to know
- Energy bar chart: KE and PE blocks at two positions
- Work diagram: force vector, displacement, angle θ
- Spring compression/extension diagram with x marked from natural length
Reactions or equations
U_net = ½m(v₂² − v₁²)
Note
This is the work–energy theorem; use when distance and speed are given/sought.
Conditions
All forces included in net work calculation; v in m/s, m in kg.
U_gravity + U_spring + U_friction + U_applied = ΔKE
Note
Break problem into individual force works, then sum.
Conditions
Multiple forces; sum all works (careful with signs).
Impulse–Momentum Principle
Formulas
∑F·Δt = m(v₂ − v₁) = Δ(mv)
Meaning
∑F = net impulse force (N); Δt = time interval (s); m·v = linear momentum (kg·m/s)
Watch out
Impulse is a vector; use signed velocities along a chosen positive axis. ∑F is NET force (all forces included).
When to use
When time interval and force (or velocity change) are involved. Best for collisions, impact, and sudden forces.
J = F_avg·Δt (impulse)
Meaning
J = impulse (N·s); F_avg = average force during interval Δt
Watch out
Impulse = change in momentum, not force alone. Do not confuse J with energy.
When to use
When force is not constant; use average force.
p = m·v (momentum)
Meaning
p = linear momentum (kg·m/s); m = mass (kg); v = velocity (m/s, signed)
Watch out
Momentum is a vector; direction matters. Use signed notation (positive/negative along chosen axis).
When to use
Define before collision problems.
Common values
Value
1 kg·m/s
Symbol
N·s or kg·m/s
Quantity
1 N·s (impulse unit)
Important facts
- Impulse–momentum avoids integration; use when Δt and velocity change are known.
- Momentum is conserved in collisions if no external horizontal impulse acts (ignore friction in short collision).
- Average force in collision: F_avg = m·Δv / Δt (mass times velocity change per unit time).
- External forces (like friction over long times) CAN alter momentum; only use conservation if external impulse ≈ 0.
- All velocities in momentum equations MUST be signed along the same positive axis; head-on collisions are sign errors waiting to happen.
Key definitions
Linear Momentum (p)
Example
A 1500 kg car at 20 m/s has p = 30,000 kg·m/s forward.
Product of mass and velocity: p = mv (kg·m/s); vector quantity.
Impulse (J)
Example
A 100 N force for 5 s gives impulse = 500 N·s.
Product of force and time: J = F·Δt (N·s); equals change in momentum.
Momentum Conservation
Example
Two colliding objects: m₁v₁ + m₂v₂ before = m₁v₁' + m₂v₂' after (no external impulse).
If no external impulse acts on a system, total momentum is constant: ∑m·v = constant.
Diagrams to know
- Momentum vector diagram: before and after collision with velocity directions
- Impulse–time graph: force vs. time, area under curve = impulse
- Collision sequence: object approach, contact, separation
Reactions or equations
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂' (no external impulse)
Note
Use signed velocities. Check units (all kg and m/s).
Conditions
Collision/impact with negligible external forces during the event.
Impact & Coefficient of Restitution
Formulas
e = (v₂ − v₁) / (u₁ − u₂)
Meaning
e = coefficient of restitution (0 ≤ e ≤ 1); u₁, u₂ = velocities BEFORE collision (signed); v₁, v₂ = velocities AFTER (signed)
Watch out
CRITICAL: Use SIGNED velocities along one common positive axis. e = (separation speed after) / (approach speed before). Head-on errors come from forgetting signs.
When to use
Any collision problem. Relates relative velocity of separation to relative velocity of approach.
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Meaning
Momentum conservation during collision (u = before, v = after); ALL velocities SIGNED.
Watch out
If two equations (momentum + restitution) and two unknowns (v₁, v₂), solve simultaneously.
When to use
Every collision problem; apply before using e.
KE_before = ½m₁u₁² + ½m₂u₂²; KE_after = ½m₁v₁² + ½m₂v₂²
Meaning
Kinetic energy values before and after; use squared speeds (always positive).
Watch out
In inelastic collision, KE_after < KE_before (energy lost to deformation, heat, sound). Do NOT assume KE is conserved unless e = 1.
When to use
Check energy loss: ΔKE = KE_after − KE_before (negative in inelastic collision).
Common values
Value
e ≈ 0.8–0.9
Symbol
e
Quantity
Steel ball on steel plate
Value
e ≈ 0.4–0.5
Symbol
e
Quantity
Wood on wood
Value
e ≈ 0.5–0.7
Symbol
e
Quantity
Rubber ball on concrete
Important facts
- Momentum is ALWAYS conserved in a collision (no external impulse during impact).
- Kinetic energy is conserved only if e = 1 (elastic). For e < 1, energy is lost.
- e = 0 (plastic): final velocities equal (objects move together). One equation for one unknown.
- e = 1 (elastic) with equal masses and head-on: objects exchange velocities (v₁' = u₂, v₂' = u₁).
- ALL velocities in e formula and momentum conservation MUST use signs (+ or −) along chosen positive axis.
- For dropped ball, e = √(rebound height / drop height) — do NOT use e = v / u directly; e = √(h₂/h₁).
Key definitions
Coefficient of Restitution (e)
Example
Ball dropped from 1 m rebounds to 0.64 m → e = √(0.64/1.0) = 0.8.
Ratio of relative separation speed to relative approach speed; 0 ≤ e ≤ 1.
Perfectly Elastic Collision (e = 1)
Example
Momentum AND energy both conserved; two equations solve for two unknowns.
Kinetic energy conserved: KE_before = KE_after. Example: billiard ball collision.
Perfectly Plastic Collision (e = 0)
Example
Truck and car collide and move together; find common velocity from momentum alone.
Objects stick together after collision; maximum energy loss. v₁' = v₂' (common final velocity).
Head-on Collision
Example
Car at +20 m/s hits stationary object (0 m/s); use +20, not just 20.
Objects move along the same line; use signed velocities (positive and negative).
Diagrams to know
- Before-and-after momentum vectors (head-on: opposite signs)
- Coefficient of restitution scale: e = 0 (plastic) to e = 1 (elastic)
- Ball drop and rebound: initial height and rebound height for e calculation
Reactions or equations
m₁u₁ + m₂u₂ = (m₁ + m₂)v_common [e = 0]
Note
One equation, one unknown; momentum conservation only.
Conditions
Perfectly plastic (objects stick); common final velocity.
e = (separation speed after) / (approach speed before) = (v₂ − v₁) / (u₁ − u₂)
Note
Two equations (momentum + e), two unknowns (v₁, v₂).
Conditions
Simultaneous with momentum conservation for elastic or intermediate e.
Curvilinear Kinetics (Circular & Curved Paths)
Formulas
∑F_n = m(v²/r) (normal/centripetal)
Meaning
∑F_n = net force toward center of curve (N); v = speed (m/s); r = radius of curve (m)
Watch out
F_n is directed toward the CENTER (inward). On a banked curve or loop, include all components (normal force, friction, weight) that have components toward center.
When to use
Any body on a curved path: banked curves, loops, circular motion. Governs the centripetal acceleration.
∑F_t = m·a_t (tangential)
Meaning
∑F_t = net force along the path direction (N); a_t = tangential acceleration (m/s²)
Watch out
Tangential is SEPARATE from centripetal. Both can act simultaneously.
When to use
Speed is changing; find acceleration along the path (e.g., speeding up on a curve).
a_n = v²/r (centripetal acceleration)
Meaning
a_n = acceleration toward center; always present on any curve, even if speed is constant.
Watch out
Centripetal acceleration is NOT zero even if tangential acceleration is zero (constant speed on curve still needs centripetal acceleration).
When to use
Before applying ∑F_n = m·a_n; calculate centripetal acceleration first.
tan(θ_bank) = v² / (rg) [ideal banking, no friction]
Meaning
θ_bank = banking angle; v = speed (m/s); r = curve radius (m); g = 9.81 m/s²
Watch out
This is the ideal angle for ONE specific speed. If speed differs, friction is needed. Do NOT assume friction = 0 if actual speed ≠ ideal speed.
When to use
Road or track banked at angle θ where car can travel at speed v without friction.
μ_min = tan(θ) − (v²/(rg·cos(θ))) / (1 + tan(θ)·v²/(rg)) [banked curve with friction]
Meaning
Minimum friction to prevent skidding on banked curve at actual speed v.
Watch out
Complex; often PRC exam asks simpler: if banked but speed not ideal, use ∑F_n = m·v²/r directly (resolve normal, friction, weight components).
When to use
Car on real banked road at speed v; find minimum μ.
Common values
Value
v_crit = √(gr); minimum speed to maintain contact at top
Symbol
v
Quantity
Critical speed (vertical loop)
Important facts
- Centripetal force ALWAYS points toward center of the curve; opposite to centrifugal effect (inertia) felt by occupants.
- On a flat curve: friction alone provides centripetal force. F_friction ≥ m·v²/r to avoid skidding.
- On a banked curve: gravity component and normal force component together provide centripetal force.
- Ideal banking angle θ: car can travel at one specific speed v with ZERO friction (∑F_n = N·sin(θ) = m·v²/r and ∑F_vertical = N·cos(θ) = mg).
- If actual speed ≠ ideal speed on banked curve, friction is needed (inward if speed too slow, outward if speed too fast).
- Vertical loop: at top of loop, normal force N and weight mg both point toward center; ∑F_n = N + mg = m·v²/r. If v too slow, N = 0 and object falls.
Key definitions
Centripetal Force (F_c)
Example
On a flat circular track, friction provides centripetal force: μ·N = m·v²/r.
Net force directed toward center of curved path; causes centripetal acceleration v²/r.
Centripetal Acceleration (a_n)
Example
Car at 10 m/s on 50 m radius curve: a_n = 100/50 = 2 m/s² inward.
Acceleration toward center: a_n = v²/r (m/s²); always perpendicular to velocity.
Banked Curve
Example
Freeway overpass banked at 10° allows car to turn without heavy friction.
Road or track tilted inward at angle θ; combines gravity and normal force to provide centripetal force.
Normal Force (N) on Curve
Example
On a banked curve, N is NOT vertical; N·sin(θ) points toward center.
Force perpendicular to surface; on banked curve, has inward (centripetal) and upward components.
Diagrams to know
- Free body diagram on flat curve: weight down, normal up, friction inward toward center
- Free body diagram on banked curve: weight down, normal at angle, components perpendicular and toward center
- Vertical loop: at top (N and mg both down/inward) and at bottom (N up, mg down, net upward)
Reactions or equations
∑F_n = m(v²/r); N·sin(θ) = m(v²/r) [banked, no friction]
Note
Also: N·cos(θ) = mg (vertical equilibrium).
Conditions
Car on banked curve at ideal speed; resolve normal force component.
N − mg·cos(θ) = m(v²/r) [vertical loop at top]
Note
Critical speed: if N = 0, then mg = m·v_crit²/r → v_crit = √(gr).
Conditions
Object at top of vertical loop; normal force and weight both inward.
Which Principle to Use: Decision Tree
Important facts
- If problem gives or asks for ACCELERATION → use ∑F = ma (Newton's 2nd law).
- If problem gives or asks for SPEED and DISTANCE → use work–energy.
- If problem gives or asks for TIME and VELOCITY CHANGE → use impulse–momentum.
- If problem is a COLLISION → use momentum conservation (impulse–momentum) + coefficient of restitution.
- If motion is on a CURVED PATH → add centripetal equation ∑F_n = m·v²/r.
Key definitions
Newton's Second Law (∑F = ma)
Example
Block on incline with friction — find acceleration → ∑F = ma.
Use when forces and acceleration are central to the problem. Requires force analysis or FBD.
Work–Energy (U = ΔKE)
Example
Push force applied over distance; find final speed → work–energy.
Use when the unknown is a speed or kinetic energy, and distance (or work path) is known. Avoids time.
Impulse–Momentum (∑F·Δt = Δ(mv))
Example
Collision, impact, or sudden force → impulse–momentum.
Use when force and time are involved, or when finding velocity change from force over time. Best for collisions.
Must remember
- ∑F = ma uses MASS (kg), not weight (N). Weight W = mg is a force; do NOT substitute into ∑F = ma without care.
- Friction ALWAYS opposes motion: F_friction = −μN (negative work). Do not add it as positive; check sign on FBD.
- Work–Energy: U_net = ½m(v₂² − v₁²). Net work equals change in kinetic energy; sum all force works (positive and negative).
- Coefficient of restitution: e = (v₂ − v₁)/(u₁ − u₂). Use SIGNED velocities along one common axis. Head-on collisions: opposite signs before → check signs.
- Momentum is ALWAYS conserved in collisions (no external impulse during impact). KE is conserved only if e = 1.
- Centripetal force ∑F_n = m(v²/r) points TOWARD center. On a banked curve, resolve normal force and gravity components inward.
- Ideal banking angle tan(θ) = v²/(rg) is for ONE speed. If actual speed differs, friction is needed.
- On a vertical loop at the top: N + mg = m(v²/r) (both inward). Critical speed: v = √(gr) when N = 0.
- Choose the right tool: acceleration → ∑F = ma; speed and distance → work–energy; time and velocity change → impulse–momentum.
- Draw a Free Body Diagram (FBD) for EVERY Newton's 2nd law problem. Label all forces, resolve along motion direction, then solve.
Last minute tips
- READ THE QUESTION CAREFULLY: Is it asking for acceleration, speed, force, or time? This tells you which principle to use immediately (∑F = ma, work–energy, or impulse–momentum).
- COLLISION PROBLEMS: Write momentum conservation FIRST; then use e formula if collision is not perfectly plastic. Always use SIGNED velocities (−10 m/s, not '10 m/s opposite'). Sign errors are the #1 mistake.
- FRICTION WORK: Always U_friction = −μNd (NEGATIVE). If you see +μNd in your work, STOP and fix it. Friction opposes motion.
- BANKED CURVE: Resolve normal force into components: N·sin(θ) inward (centripetal), N·cos(θ) upward. If problem says 'ideal banking,' then tan(θ) = v²/(rg) (friction = 0).
- VERTICAL LOOP AT TOP: Remember N + mg = m(v²/r), not N − mg. Both point toward center (downward). If N comes out negative, that's physically impossible (object loses contact).
Comparison tables
Collisions: Elastic vs. Plastic
| Property | Perfectly Elastic (e = 1) | Perfectly Plastic (e = 0) | Partially Elastic (0 < e < 1) |
|---|---|---|---|
| Objects after collision | Separate with different velocities | Stick together (common velocity) | Separate with intermediate velocities |
| Momentum conservation | Yes | Yes | Yes |
| Kinetic energy conservation | Yes (KE before = KE after) | No (KE lost) | No (KE lost) |
| Energy loss | None | Maximum (deformation, heat) | Some (partial deformation) |
| Number of equations needed | 2 (momentum + energy) | 1 (momentum only; v₁' = v₂') | 2 (momentum + e formula) |
| Typical example | Billiard balls, steel spheres | Truck crash (vehicles crumple) | Most real collisions |
Three Kinetics Principles: When to Use
| Principle | Equation | When to Use | Best For | Weakness |
|---|---|---|---|---|
| Newton's 2nd Law | ∑F = ma | Forces & acceleration known/sought | Direct force analysis, FBD-based | Requires FBD; must find all forces |
| Work–Energy | U = ½m(v₂² − v₁²) | Distance & speed known; time not involved | Speed calculation over distance | Cannot find time or acceleration directly |
| Impulse–Momentum | ∑F·Δt = m(v₂ − v₁) | Time interval & force/velocity change involved | Collisions, impact, sudden forces | Requires time interval; less useful for steady motion |
Signs in Momentum & Collision Problems (CRITICAL)
| Scenario | Positive Direction | Mass 1 Velocity Before | Mass 2 Velocity Before | Sign Convention |
|---|---|---|---|---|
| Head-on collision (car A hits car B head-on) | Choose A's direction as + | v₁ = +20 m/s (to the right) | v₂ = −15 m/s (to the left) | ALWAYS use signed velocities; negative means opposite direction |
| Rear-end collision (car A hits car B from behind, same direction) | Forward is + | v₁ = +25 m/s (faster) | v₂ = +15 m/s (slower, same direction) | Both positive; A catches B |
| Ball dropped vertically | Down is + | v_before = +√(2gh) (downward) | v_after = −e·√(2gh) (upward after bounce) | Velocity after reverses sign; use e = (|v_after| / |v_before|) or signed definition |
Diagrams
Kinetics Problem Flowchart: Which Principle to Use
Decision tree to rapidly identify whether to use Newton's 2nd law, work–energy, impulse–momentum, or centripetal principles. Follow from the primary given/sought quantity.
Kinetics: Key Formulas & Concepts Map
Mind map showing the four main kinetics principles and their key formulas/concepts. Use this as a rapid reference to see what falls under each principle.
Collision Outcome States Based on e
State diagram showing collision outcomes based on the coefficient of restitution value. Clarifies which analysis method and energy assumption to use.