8–1 Description of Motion
Record the position of a point as a function of time, using functions or graphs: s=f(t). Example: free fall s=16t² (imperial units), the curve is a parabola.
We start with a one-dimensional approximation, ignoring microscopic quantum and relativistic details, focusing on the basic quantification of macroscopic motion.
8–2 Velocity: The Limiting Definition and the Birth of Calculus
Instantaneous velocity is the limit of "displacement/time over an extremely short interval":
v = lim_{Δt→0} Δs/Δt = ds/dt
Originated from Zeno's paradoxes and the difficulty of defining speed; Newton and Leibniz developed differential calculus.
8–3 Derivative Notation and Differentiation Examples
If s=16t², then v=ds/dt=32t. General function s=At³+Bt+C ⇒ v=3At²+B.
Common differentiation rules: sum, constant multiple, power function, product rule, etc.
8–4 Distance as an Integral
Given a velocity table v(t), the distance traveled is the integral of velocity over time:
s = ∫ v(t) dt
Integration is the limit of summation; numerical integration approximates by accumulating over short time slices.
8–5 Acceleration and Three-Dimensional Motion
Acceleration a=dv/dt=d²s/dt². Constant acceleration free fall: v=gt, s=½gt².
v = (v_x²+v_y²+v_z²)^{1/2}; v_x=dx/dt, a_x=dv_x/dt
Compound motion: horizontal uniform velocity + vertical constant acceleration ⇒ parabola y=−(g/2u²)x².
Key Points
- Position-time functions describe motion; graphs and tables correspond to each other.
- Instantaneous velocity is defined by limits; calculus provides the language of kinematics.
- Integration recovers displacement from velocity; acceleration is the derivative of velocity.
- 2D/3D component method: decompose and compose v, a; projectile motion is a typical example.