Dynamics: Kinematics - Cheat Sheet
Your last-minute revision companion for kinematics: rectilinear motion, projectile motion, and rotational kinematics. All formulas, conversions, and exam pitfalls condensed for rapid recall in the final 30 minutes.
Rectilinear Motion — Constant Acceleration
Formulas
v = v₀ + at
Meaning
v = final velocity (m/s); v₀ = initial velocity (m/s); a = acceleration (m/s²); t = time (s)
Watch out
Sign of a: negative means deceleration (slowing down). Be consistent with your positive direction.
When to use
When finding velocity after a known time interval with constant acceleration.
s = v₀t + ½at²
Meaning
s = displacement (m); v₀ = initial velocity (m/s); a = acceleration (m/s²); t = time (s)
Watch out
This is DISPLACEMENT, not distance. Direction matters. If motion reverses, use calculus.
When to use
When finding displacement over a time interval; use this when time is known.
v² = v₀² + 2as
Meaning
v = final velocity (m/s); v₀ = initial velocity (m/s); a = acceleration (m/s²); s = displacement (m)
Watch out
This equation works only for CONSTANT acceleration. Cannot directly find v if a varies with position.
When to use
When you know displacement and acceleration but NOT time. Eliminates t from the problem.
a = (v − v₀)/t = Δv/Δt
Meaning
Average acceleration is the change in velocity divided by time elapsed.
Watch out
For variable acceleration, this gives AVERAGE a, not instantaneous a. Use calculus for instantaneous.
When to use
Rearrange to find time or initial velocity when acceleration is constant.
Common values
Value
9.81 m/s² or 9.80 m/s² (often 10 m/s² for approximation)
Symbol
g
Quantity
Acceleration due to gravity (standard)
Important facts
- The three kinematic equations (v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as) are interrelated; each omits one variable.
- Constant acceleration is the only case where these simple formulas apply directly; variable acceleration requires integration.
- For free fall, use a = g = 9.81 m/s² (downward is positive if you define downward as +y).
- v–t graphs: slope = acceleration; area under curve = displacement.
- s–t graphs: slope = velocity; curvature indicates acceleration (parabolic for constant a).
Key definitions
Displacement (s)
Example
Car moves 50 m east then 20 m west: displacement = 30 m east; distance = 70 m.
Vector quantity representing the change in position; can be negative (opposite direction).
Velocity (v)
Example
v = ds/dt; instantaneous velocity at t = 2 s is the slope of the s–t curve at that point.
Rate of change of displacement with respect to time; includes direction (vector).
Acceleration (a)
Example
Constant deceleration of −4 m/s² means velocity decreases by 4 m/s every second.
Rate of change of velocity with respect to time; a = dv/dt = v(dv/ds).
Deceleration
Example
Car braking: a = −4 m/s² (if forward is positive).
Acceleration in the direction opposite to velocity; represented as negative acceleration.
Diagrams to know
- v–t (velocity vs. time) graph for constant acceleration: straight line with slope = a.
- s–t graph for constant acceleration: parabola (upward if a > 0, downward if a < 0).
- a–t graph for constant acceleration: horizontal line at constant value.
Rectilinear Motion — Variable Acceleration
Formulas
v = ∫a dt
Meaning
Velocity is the integral of acceleration with respect to time; add constant of integration.
Watch out
Must know the initial condition (v at t = 0) to evaluate the constant of integration.
When to use
When acceleration is a function of time: a = a(t).
s = ∫v dt
Meaning
Displacement is the integral of velocity with respect to time.
Watch out
Two levels of integration: first a → v, then v → s. Each adds an integration constant.
When to use
After finding v(t), integrate to get s(t). Again, use initial position s₀.
a = v(dv/ds)
Meaning
Acceleration expressed as a function of position: use when a = a(s).
Watch out
This is an alternative form of a = dv/dt; rearrange and separate variables to solve.
When to use
When acceleration depends on position, not time. Relates a, v, and s directly.
Important facts
- When a is given as a function of time a(t), integrate twice: v = ∫a dt, then s = ∫v dt.
- When a is given as a function of position a(s), use a = v(dv/ds) and separate variables.
- Always evaluate constants of integration using initial conditions (v₀ at t = 0, s₀ at t = 0).
- Board exams rarely give complex variable-a problems; most are constant-a disguised as variable.
Key definitions
Variable acceleration
Example
a = 2t (acceleration increases linearly with time); a = kv (air resistance proportional to velocity).
Acceleration that changes with time, position, or velocity; requires calculus to solve.
Diagrams to know
- Curved v–t graph: slope at any point = instantaneous acceleration a(t).
- Area under a–t graph from t₁ to t₂ = change in velocity Δv.
Free Fall and Vertical Motion
Formulas
v = gt (or v = v₀ + gt for non-zero initial velocity)
Meaning
v = velocity at time t; g = 9.81 m/s² (downward positive); v₀ = initial velocity.
Watch out
Sign convention: define downward as + or −, then be consistent. Upward throws start with −v₀.
When to use
Object dropped or thrown vertically; assume no air resistance.
h = v₀t + ½gt²
Meaning
h = vertical height fallen (m); v₀ = initial upward velocity (m/s); t = time (s); g = 9.81 m/s².
Watch out
If thrown upward, v₀ > 0 and h decreases initially; ½gt² always opposes the chosen direction.
When to use
Finding height or depth after time t in vertical motion.
v² = v₀² + 2gh
Meaning
Final velocity squared in terms of height fallen; links velocity and height without time.
Watch out
If thrown upward, v₀ is positive; at max height, v = 0 and h = h_max.
When to use
When time is unknown; most common in impact-speed problems.
t_total = (2v₀)/g (object thrown upward, returns to starting height)
Meaning
Time of flight for symmetric projectile launched and landing at same height.
Watch out
Applies ONLY if launch and landing heights are equal. Different heights: solve h = 0 directly.
When to use
Vertical launch with upward initial velocity v₀.
Common values
Value
14.0 m/s
Symbol
v = √(2·9.81·10)
Quantity
Impact speed from 10 m height (dropped from rest)
Value
1.43 s
Symbol
t = √(2·10/9.81)
Quantity
Time to fall 10 m from rest
Important facts
- At maximum height, v = 0; use v² = v₀² + 2gh or v = v₀ − gt to find time to max height.
- On return to starting height, final speed equals initial speed (by energy symmetry); direction is opposite.
- Air resistance is ALWAYS ignored in kinematics unless explicitly stated.
- Impact speed from height h (dropped from rest): v = √(2gh). For h = 10 m, g = 9.81, v ≈ 14 m/s.
Key definitions
Free fall
Example
Stone dropped from cliff or ball thrown upward (both experience g throughout motion).
Motion under gravity alone, with no air resistance; acceleration = g = 9.81 m/s² downward.
Diagrams to know
- Vertical motion graphs: v–t is a straight line (slope = −g if upward is +); s–t is inverted parabola.
Projectile Motion
Formulas
x = (v₀ cos θ)t
Meaning
Horizontal displacement; v₀ = launch speed (m/s); θ = launch angle; t = time (s).
Watch out
Horizontal velocity NEVER changes (no air resistance). This is NOT accelerated motion.
When to use
Horizontal motion is uniform (constant velocity); find how far projectile travels horizontally.
y = (v₀ sin θ)t − ½gt²
Meaning
Vertical displacement above launch point; second term is gravity effect.
Watch out
At landing, y = 0 (if landing at same height); solve for t, then find range via x = (v₀ cos θ)t.
When to use
Vertical motion includes gravity; describe height at any time t.
R = (v₀² sin 2θ)/g
Meaning
Range (horizontal distance): applies when projectile lands at same height as launch.
Watch out
Uses sin 2θ, NOT sin² θ. Maximum range at θ = 45°. Complementary angles (e.g., 30° and 60°) give same range.
When to use
Find how far a projectile travels horizontally (landing height = launch height).
H = (v₀² sin² θ)/(2g)
Meaning
Maximum height above launch point; vertical component only.
Watch out
Uses sin² θ (different from range formula). At max height, v_y = 0; time to max height = (v₀ sin θ)/g.
When to use
Find peak height of trajectory.
t_flight = (2v₀ sin θ)/g
Meaning
Total time in air for symmetric trajectory (landing at same height).
Watch out
This is exactly twice the time to reach max height: t_max = (v₀ sin θ)/g.
When to use
Find how long projectile is airborne.
v_x = v₀ cos θ (constant)
Meaning
Horizontal component of velocity remains constant throughout flight.
Watch out
ALWAYS constant. Vertical component v_y changes due to gravity.
When to use
Analyze velocity at any instant; horizontal component never changes.
v_y = v₀ sin θ − gt
Meaning
Vertical component of velocity at time t; decreases due to gravity.
Watch out
Positive upward. At landing (if same height), v_y = −v₀ sin θ (downward, same magnitude as launch).
When to use
Find vertical velocity at any time; becomes zero at max height.
v = √(v_x² + v_y²)
Meaning
Magnitude of total velocity at any instant; combine components vectorially.
Watch out
Speed is scalar (magnitude only). Use Pythagoras to combine perpendicular components.
When to use
Find speed (not velocity component) at a given time or position.
Common values
Value
91.7 m
Symbol
R = (900 × sin 90°)/9.81
Quantity
Range at 45° launch with v₀ = 30 m/s
Value
22.9 m
Symbol
H = (900 × sin² 45°)/(2 × 9.81)
Quantity
Max height at 45° launch with v₀ = 30 m/s
Value
4.33 s
Symbol
t = (2 × 30 × sin 45°)/9.81
Quantity
Flight time at 45° launch with v₀ = 30 m/s
Important facts
- Horizontal and vertical motions are INDEPENDENT: solve them separately and combine results via time t.
- Maximum range occurs at θ = 45° for symmetric trajectories: R_max = v₀²/g.
- Complementary angles θ and (90° − θ) give the same range but different max heights and flight times.
- At max height, v_y = 0 but v_x ≠ 0; speed is not zero, only vertical component is.
- Landing speed (if landing at same height) has same magnitude as launch speed but different direction.
- If landing height ≠ launch height, solve y = (v₀ sin θ)t − ½gt² for t when y = height difference, then find x.
Key definitions
Projectile
Example
Thrown ball, launched rocket, or water from a fountain; all follow parabolic paths.
Object moving under gravity alone, with initial velocity at an angle; horizontal and vertical motions are independent.
Range (R)
Example
Soccer kick with v₀ = 20 m/s at 45°: R = (400 × sin 90°)/9.81 ≈ 40.8 m.
Horizontal distance traveled by projectile, from launch to landing at same height.
Maximum height (H)
Example
Ball thrown upward at 20 m/s: H = (400 × sin² 90°)/(2 × 9.81) ≈ 20.4 m (if θ = 90°).
Peak vertical displacement above the launch point; occurs when v_y = 0.
Diagrams to know
- Parabolic trajectory: symmetric if landing = launch height; longer on one side if landing lower.
- v–t components: v_x is constant horizontal line; v_y is a downward-sloping line (slope = −g).
- Velocity vector at any point: combine v_x (horizontal) and v_y (vertical) components.
Rotational (Angular) Kinematics
Formulas
ω = ω₀ + αt
Meaning
ω = final angular velocity (rad/s); ω₀ = initial angular velocity (rad/s); α = angular acceleration (rad/s²); t = time (s).
Watch out
Must use radians, not degrees or revolutions. Sign of α: negative means deceleration (slowing rotation).
When to use
Constant angular acceleration; find angular velocity after known time.
θ = ω₀t + ½αt²
Meaning
θ = angular displacement (rad); ω₀ = initial angular velocity (rad/s); α = angular acceleration (rad/s²); t = time (s).
Watch out
θ is in RADIANS. Convert revolutions: 1 rev = 2π rad. If α varies, integrate ω instead.
When to use
Constant angular acceleration; find angular displacement over time interval.
ω² = ω₀² + 2αθ
Meaning
Final angular velocity squared in terms of angular displacement; eliminates time.
Watch out
Again, use radians for θ. This is the rotational analogue of v² = v₀² + 2as.
When to use
When time is unknown; links angular velocity and displacement.
α = (ω − ω₀)/t
Meaning
Angular acceleration is the change in angular velocity per unit time.
Watch out
For constant α. If α varies with time or position, use calculus (α = dω/dt or α = ω dω/dθ).
When to use
Solve for α, ω₀, or t when others are known.
Common values
Value
ω (rad/s) = (2π × rpm)/60 ≈ 0.10472 × rpm
Symbol
rpm ↔ rad/s
Quantity
Conversion: revolutions per minute to radians per second
Value
31.42 rad/s
Symbol
ω = (2π × 300)/60
Quantity
300 rpm in rad/s
Important facts
- Rotational equations mirror rectilinear ones: (θ, ω, α) ↔ (s, v, a); use same three-equation set.
- ALWAYS convert rpm or Hz to rad/s before using kinematic equations: ω (rad/s) = (2π × N)/60 where N is rpm.
- Revolutions: 1 rev = 2π rad. To find revolutions, calculate θ in radians and divide by 2π.
- Angular velocity and acceleration can be positive (counterclockwise) or negative (clockwise); define positive direction first.
- Constant angular acceleration is common in rotors, motors, and flywheels; variable α requires integration.
Key definitions
Angular velocity (ω)
Example
Flywheel rotating at 300 rpm ≡ ω = (2π × 300)/60 = 31.42 rad/s.
Rate of change of angular position (angle) with respect to time; ω = dθ/dt (rad/s).
Angular acceleration (α)
Example
Motor accelerating from 0 to 1500 rpm in 5 s: α ≈ 31.4 rad/s² (convert rpm to rad/s first).
Rate of change of angular velocity with respect to time; α = dω/dt (rad/s²).
Radian (rad)
Example
A wheel rotating 2 revolutions = 4π rad ≈ 12.57 rad.
Unit of angle in kinematics; 1 rev = 2π rad ≈ 6.283 rad; 1 rad ≈ 57.3°.
Diagrams to know
- ω–t (angular velocity vs. time) graph: straight line for constant α (slope = α).
- θ–t graph: parabola for constant α; curvature indicates direction and magnitude of α.
Linking Rotation and Linear Motion
Formulas
s = rθ
Meaning
Arc length s (m) traveled by a point at radius r (m) through angle θ (rad).
Watch out
θ MUST be in radians, not degrees. If θ is in degrees, convert: θ_rad = θ_deg × π/180.
When to use
Convert angular displacement to arc length for a point on a rotating object.
v = rω
Meaning
Linear (tangential) velocity v (m/s) of a point at radius r (m) rotating with angular velocity ω (rad/s).
Watch out
ω must be in rad/s. This is the TANGENTIAL velocity (along the circle), not radial.
When to use
Convert angular velocity to linear speed at any radius; essential for gears, pulleys, wheels.
a_t = rα
Meaning
Tangential acceleration a_t (m/s²) at radius r due to angular acceleration α (rad/s²).
Watch out
This is TANGENTIAL (along motion), not the centripetal acceleration a_n = v²/r pointing inward.
When to use
Find the linear acceleration in the direction of motion (along the circle).
a_n = v²/r = rω²
Meaning
Normal (centripetal) acceleration a_n (m/s²); directed toward the center; always present in circular motion.
Watch out
This is NOT zero in circular motion; it exists even at constant ω. Use v²/r if you know linear speed, rω² if angular speed.
When to use
Find radial acceleration (center-pointing); use either v²/r or rω² depending on what you know.
a_total = √(a_t² + a_n²)
Meaning
Total acceleration is vector sum of tangential and normal (centripetal) components.
Watch out
These are perpendicular; use Pythagoras. Do not add magnitudes directly.
When to use
Find magnitude of total acceleration when both tangential and centripetal components are present.
Common values
Value
20 m/s²
Symbol
a_n = 100/5
Quantity
Centripetal acceleration for v = 10 m/s, r = 5 m
Important facts
- For a point on a rotating body: v = rω (tangential), a_n = rω² (centripetal), a_t = rα (tangential rate of change).
- Centripetal acceleration a_n is ALWAYS present and directed INWARD (toward axis) in any circular motion.
- Tangential acceleration a_t is zero if ω is constant (uniform circular motion); nonzero if ω is changing (nonuniform).
- Different points on the same rotating object have different linear speeds v and accelerations a, but same ω and α.
- In gears or pulleys: the tangential velocity is the same at the meshing points, so v₁ = v₂, thus r₁ω₁ = r₂ω₂.
Key definitions
Tangential velocity
Example
Rim of a wheel 0.5 m radius rotating at 20 rad/s: v = 0.5 × 20 = 10 m/s (tangent to wheel).
Linear velocity of a point on a rotating body, directed along the circle (perpendicular to radius); v = rω.
Centripetal (normal) acceleration
Example
Car on a circular track at 15 m/s with radius 50 m: a_n = 225/50 = 4.5 m/s² (toward center).
Acceleration directed toward the center of rotation; a_n = v²/r = rω²; always nonzero in circular motion.
Tangential acceleration
Example
Wheel accelerating at α = 2 rad/s² at radius 0.3 m: a_t = 0.3 × 2 = 0.6 m/s² (along rim).
Component of acceleration along the direction of motion; a_t = rα; nonzero only if ω is changing.
Diagrams to know
- Rotating body: velocity v is tangent to circle; centripetal a_n points inward; tangential a_t is along velocity direction.
- Free-body diagram of rotating point: show centripetal acceleration (inward) and tangential acceleration (along motion).
Unit Conversions in Kinematics
Common values
Value
2π/60 ≈ 0.10472
Symbol
ω (rad/s) = N × (2π/60) where N is rpm
Quantity
Conversion factor: rpm to rad/s
Value
60/(2π) ≈ 9.549
Symbol
N (rpm) = ω × 60/(2π)
Quantity
Conversion factor: rad/s to rpm
Important facts
- Angular velocity: 1 rpm = (2π/60) rad/s ≈ 0.10472 rad/s; conversely, 1 rad/s ≈ 9.549 rpm.
- Angular position: 1 rev = 2π rad = 360°; 1 rad ≈ 57.3°; 1° = π/180 rad ≈ 0.01745 rad.
- Frequency: 1 Hz = 1 rev/s = 60 rpm = 2π rad/s.
- Time: 1 min = 60 s; hours to seconds and vice versa are rare in kinematics but common in real applications.
- Speed: 1 m/s = 3.6 km/h; conversely, 1 km/h ≈ 0.278 m/s.
Must remember
- The three constant-acceleration kinematic equations are ALWAYS linked; if you know any two of (v, s, t, a), you can find the third using the appropriate equation.
- For projectile motion, horizontal motion (x = v₀ cos θ · t) is INDEPENDENT of vertical motion (y = v₀ sin θ · t − ½gt²); solve separately, combine via time t.
- Maximum range in projectile motion occurs at θ = 45° and equals R_max = v₀²/g; complementary angles (e.g., 30° and 60°) yield the same range but different flight times and heights.
- Angular equations (ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ) mirror linear equations exactly; θ ↔ s, ω ↔ v, α ↔ a.
- ALWAYS convert rpm to rad/s using ω = (2π N)/60 before plugging into rotational kinematic equations; units are critical.
- Centripetal (normal) acceleration a_n = v²/r = rω² ALWAYS points toward the center and is ALWAYS nonzero in any circular motion, even if ω is constant.
- Displacement (vector) and distance (scalar) are different; in kinematics, use displacement s, which can be negative; solve for time first, then check if motion reverses.
- Free fall or vertical motion: define downward as + or −, then apply g = 9.81 m/s² consistently; upward throws have negative initial velocity under this convention.
- For variable acceleration, use integration: v = ∫a dt and s = ∫v dt; always evaluate constants of integration using initial conditions (v₀ and s₀).
- In circular motion of a point: tangential velocity v = rω, tangential acceleration a_t = rα (nonzero only if α ≠ 0), centripetal acceleration a_n = rω² (always nonzero).
Last minute tips
- Double-check unit conversions: rpm → rad/s, degrees → radians. A single unit error cascades through the entire solution. Always state your conversion factor explicitly.
- For projectile problems, always identify whether launch and landing heights are equal; if not, solve y = (v₀ sin θ)t − ½gt² = Δy for t before finding range. Do NOT use the symmetric-trajectory formulas blindly.
- In free-fall problems, consistently define your positive direction (up or down) at the START. Then apply g with the correct sign throughout. Sign errors are the #1 mistake on exams.
- For rotational kinematics, always ask: 'Is ω constant or varying?' If constant, a_n = rω² and a_t = 0. If ω is changing, calculate both a_t = rα and a_n = rω² and find total via Pythagoras.
- On the exam, use s = v₀t + ½at² or v² = v₀² + 2as depending on what you're given; these are faster than integrating variable acceleration. Save calculus for questions that explicitly give a(t) or a(s).
Comparison tables
Rectilinear vs. Rotational Kinematics (Constant Acceleration)
| Quantity | Rectilinear (Linear) | Rotational (Angular) | Link |
|---|---|---|---|
| Position/Displacement | s (m) | θ (rad) | s = rθ |
| Velocity | v (m/s) | ω (rad/s) | v = rω |
| Acceleration | a (m/s²) | α (rad/s²) | a_t = rα (tangential) |
| 1st Equation | v = v₀ + at | ω = ω₀ + αt | Direct parallel |
| 2nd Equation | s = v₀t + ½at² | θ = ω₀t + ½αt² | Direct parallel |
| 3rd Equation | v² = v₀² + 2as | ω² = ω₀² + 2αθ | Direct parallel |
Projectile Motion: Key Formulas and Scenarios
| Scenario | Launch Height vs. Landing | Formula for Time | Formula for Range | Formula for Max Height |
|---|---|---|---|---|
| Symmetric trajectory | Same height | t = (2v₀ sin θ)/g | R = (v₀² sin 2θ)/g | H = (v₀² sin² θ)/(2g) |
| Launch from ground level | y = 0 at start and end | Solve (v₀ sin θ)t − ½gt² = 0 | Use symmetric formulas | Occurs at t = (v₀ sin θ)/g |
| Different heights | y_land ≠ y_launch | Solve y = (v₀ sin θ)t − ½gt² = Δy | x = (v₀ cos θ) × t_land | Same formula; occurs when v_y = 0 |
| Max range angle | Same height only | θ = 45° | R_max = v₀²/g | H_45° = v₀²/(4g) |
Common Mistakes in Kinematics (Watch-Out Summary)
| Mistake | Correct Approach | Why It Matters | |
|---|---|---|---|
| Forgetting to convert rpm to rad/s | Plugging rpm directly into rotational equations | Always convert: ω = 2πN/60 where N = rpm | Equations require SI units; answers will be off by ~10. |
| Using sin θ instead of sin 2θ for range | R = (v₀² sin θ)/g (WRONG) | R = (v₀² sin 2θ)/g (CORRECT) | Drastically wrong numerical answer; only sin 2θ gives correct range. |
| Confusing distance and displacement | Treating them as same in calculations | Distance is scalar (total path); displacement is vector (net change) | Sign and direction matter in rectilinear motion; affects final answer. |
| Not using radians for angular kinematics | Plugging degrees or revolutions into equations | Convert everything to radians before solving | Formulas s = rθ, v = rω derived assuming θ in radians; degree use gives wrong results. |
| Forgetting sign of acceleration in free fall | Treating deceleration (upward throw) as positive | Define direction first, then apply sign consistently | Wrong sign → wrong time to max height, wrong impact velocity. |
| Ignoring centripetal acceleration in circular motion | Assuming a = 0 when ω is constant | Centripetal a_n = v²/r = rω² always exists if there's rotation | Total acceleration is nonzero even in uniform circular motion. |
Diagrams
Kinematics Topics and Key Formulas
Comprehensive mind map showing the three main kinematics topics (rectilinear, projectile, rotational) and their key formulas. Use this to quickly locate the correct equation for your problem type.
Decision Tree: Which Kinematic Equation to Use
Flowchart to help identify which equation to use based on the motion type and known variables. Start at the top and follow branches to select the correct formula for your problem.
Projectile Motion: State Transitions During Flight
State diagram showing the phases of projectile motion from launch through peak to landing. Illustrates how vertical velocity changes while horizontal velocity remains constant, and highlights key kinematics at each state.