Kinematics - Formulas & Relationships
IB Mathematics Analysis & Approaches (SL & HL)
📐 Basic Quantities in Motion
Displacement (\(s\)):
• Change in position from starting point (vector quantity)
• Measured in meters (m)
• Can be positive, negative, or zero
Velocity (\(v\)):
• Rate of change of displacement (vector quantity)
• Measured in meters per second (m/s or ms⁻¹)
• Has both magnitude and direction
Acceleration (\(a\)):
• Rate of change of velocity (vector quantity)
• Measured in meters per second squared (m/s² or ms⁻²)
• Negative acceleration = deceleration (slowing down)
Speed vs Distance:
• Speed: Magnitude of velocity (scalar quantity)
• Distance: Total path length traveled (scalar quantity, always positive)
🔻 Differentiation (Finding Rate of Change)
Velocity from Displacement:
\[v = \frac{ds}{dt}\]
Velocity is the first derivative of displacement with respect to time
Acceleration from Velocity:
\[a = \frac{dv}{dt}\]
Acceleration is the first derivative of velocity with respect to time
Acceleration from Displacement:
\[a = \frac{d^2s}{dt^2}\]
Acceleration is the second derivative of displacement with respect to time
Summary Chain:
\[\text{Displacement} \xrightarrow{\text{differentiate}} \text{Velocity} \xrightarrow{\text{differentiate}} \text{Acceleration}\]
🔺 Integration (Finding Cumulative Change)
Velocity from Acceleration:
\[v = \int a\,dt\]
Remember to add constant of integration \(+C\)
Displacement from Velocity:
\[s = \int v\,dt\]
Remember to add constant of integration \(+C\)
Displacement Between Two Times:
\[s = \int_{t_1}^{t_2} v(t)\,dt\]
Definite integral gives net displacement (can be negative)
Summary Chain:
\[\text{Acceleration} \xrightarrow{\text{integrate}} \text{Velocity} \xrightarrow{\text{integrate}} \text{Displacement}\]
📏 Distance vs Displacement
Total Distance Traveled:
\[\text{Distance} = \int_{t_1}^{t_2} |v(t)|\,dt\]
Use absolute value to ensure distance is always positive
Key Difference:
• Displacement: Net change in position (can be negative)
• Distance: Total path length (always positive or zero)
• If direction doesn't change: Distance = |Displacement|
• If direction changes: Distance > |Displacement|
⚡ Kinematic Equations (Constant Acceleration)
Variables:
• \(s\) = displacement
• \(u\) = initial velocity
• \(v\) = final velocity
• \(a\) = acceleration (constant)
• \(t\) = time
Equation 1 (Velocity-Time):
\[v = u + at\]
Equation 2 (Displacement with Time):
\[s = ut + \frac{1}{2}at^2\]
Equation 3 (Velocity-Displacement):
\[v^2 = u^2 + 2as\]
Equation 4 (Average Velocity):
\[s = \frac{u + v}{2}t\]
🎯 Special Conditions
Particle at Rest:
\[v = 0\]
Velocity is zero (momentarily stopped)
Maximum/Minimum Displacement:
\[\frac{ds}{dt} = v = 0\]
Displacement is at maximum or minimum when velocity is zero
Changes Direction:
When velocity changes sign (from positive to negative or vice versa)
Constant Velocity:
\[a = 0\]
No acceleration means constant velocity (uniform motion)
📈 Graph Interpretations
Displacement-Time Graph:
• Gradient: Velocity
• Steeper gradient: Higher velocity
• Horizontal line: At rest (v = 0)
• Curved line: Acceleration is present
Velocity-Time Graph:
• Gradient: Acceleration
• Area under curve: Displacement
• Horizontal line: Constant velocity
• Crosses x-axis: Changes direction
Acceleration-Time Graph:
• Area under curve: Change in velocity
• Horizontal line: Constant acceleration
• Positive area: Increasing velocity
• Negative area: Decreasing velocity (deceleration)
🔢 Finding Constants of Integration
Initial Conditions:
• Phrases like "initially", "at the start", "when \(t = 0\)"
• Substitute \(t = 0\) and given values to find constant \(C\)
• Example: "Initially at rest" means \(v = 0\) when \(t = 0\)
• Example: "Starts at origin" means \(s = 0\) when \(t = 0\)
Boundary Conditions:
• Information given at a specific time (not necessarily \(t = 0\))
• Substitute the given time and values to find constant \(C\)
• Example: "When \(t = 3\), \(s = 5\)"
✅ Problem-Solving Strategy
Steps for Kinematics Problems:
1. Identify what you're given and what you need to find
2. Determine if you need to differentiate or integrate
3. Write the appropriate relationship formula
4. Perform the calculus operation
5. Use initial/boundary conditions to find constants
6. Check your answer makes physical sense
7. Include units in your final answer
Common Question Types:
• Find when particle is at rest: Set \(v = 0\), solve for \(t\)
• Find maximum displacement: Set \(v = 0\), find \(s\) at that time
• Find total distance: Integrate \(|v|\) over time interval
• Find when particle changes direction: Find when \(v\) changes sign
💡 Exam Tip: Remember the key relationships: differentiate to find rate of change (s → v → a), integrate to find cumulative change (a → v → s). For distance vs displacement, use absolute value for distance. Always check initial conditions to find constants. Use your GDC to verify integrals and derivatives. Graph sketching helps visualize motion - practice interpreting displacement, velocity, and acceleration graphs. Units matter: m for displacement, m/s for velocity, m/s² for acceleration!