Overview
This lecture reviews the core motion equations for one-dimensional kinematics, introduces their differential and integral forms, and demonstrates their application for both constant and variable acceleration.
Relationships Between Position, Velocity, and Acceleration
- Position is represented by s.
- Instantaneous velocity (v) is the first derivative of position with respect to time: v = ds/dt.
- Instantaneous acceleration (a) is the derivative of velocity with respect to time: a = dv/dt.
- Acceleration can also be written as a = v · dv/ds.
Differential and Integral Forms of Kinematic Equations
- To move from position to velocity, differentiate; to move from velocity to acceleration, differentiate again.
- To go from acceleration to velocity or position, integrate.
- Key equations:
- ∫(s₀ to s) ds = ∫(0 to t) v dt
- ∫(v₀ to v) dv = ∫(0 to t) a dt
- ∫(s₀ to s) a ds = ∫(v₀ to v) v dv
Constant Acceleration Equations
- When acceleration is constant (e.g., gravity, g = 9.81 m/s²), equations simplify:
- v = v₀ + a·t
- s = s₀ + v₀·t + ½a·t²
- v² = v₀² + 2a(s - s₀)
- These only apply if acceleration does not change with time.
Example Problem: Variable Acceleration
- Given v = 4t – 3t² (m/s), s = 0 at t = 0.
- Find acceleration at t = 4: a = dv/dt = 4 – 6t; at t = 4, a = –20 m/s².
- Find position at t = 4:
- Integrate velocity: s = ∫₀⁴ (4t – 3t²) dt = 2t² – t³ evaluated from 0 to 4 = –32 m.
- Since acceleration is not constant, must use the general integral forms, not the constant-acceleration equations.
Key Terms & Definitions
- Position (s) — The location of a particle at a given time.
- Velocity (v) — Rate of change of position with respect to time.
- Acceleration (a) — Rate of change of velocity with respect to time.
- Constant acceleration — Acceleration that does not vary over time.
- Integration — Mathematical process to find area under a curve; used to move from acceleration to velocity or position.
Action Items / Next Steps
- Practice using both general and constant-acceleration kinematic equations.
- Complete assigned problems involving both forms, especially where acceleration is not constant.