9–1 Momentum and Force
Momentum p = mv. The general form of Newton's second law:
F = d(mv)/dt; (constant mass) ⇒ F = ma
Acceleration can change either the magnitude or direction of velocity; the centripetal acceleration of uniform circular motion is a=v²/R.
9–2 Velocity and Speed
Velocity includes direction; speed is its magnitude. Three components: v_x=dx/dt, v_y=dy/dt, v_z=dz/dt; |v|=(v_x²+v_y²+v_z²)^{1/2}.
9–3 Component Form
F_x = m d²x/dt², F_y = m d²y/dt², F_z = m d²z/dt²
Force and motion can be analyzed independently in each direction; for example, horizontal uniform motion and vertical free fall are independent of each other.
9–4 Where Forces Come From: Gravity and Springs
- Near-surface gravity: F=mg ⇒ x-direction a=g, v_x=v_0+gt, x=x_0+v_0 t + ½gt².
- Spring: F=−kx ⇒ m d²x/dt² + kx=0 ⇒ simple harmonic motion x=cos(ωt), etc. (solvable numerically or analytically).
9–5/9–6 Essentials of Numerical Methods
Iterate with small step size ε: x(t+ε)=x(t)+ε v(t+ε/2); v(t+ε/2)=v(t−ε/2)+ε a(t).
The midpoint method significantly improves accuracy; examples: harmonic oscillator and orbital approximation (matching the analytical solution x=cos t).
9–7 Numerical Computation of Planetary Motion
a_x=−x/r³, a_y=−y/r³, r=(x²+y²)^{1/2}
Small-step integration can trace elliptical orbits; extend to N-body: for each pair (i,j), accumulate −G m_j (r_i−r_j)/|r_i−r_j|³.
Key Points
- The most general form of Newton's second law is F=d(mv)/dt.
- The component method handles 3D motion with independent superposition.
- Harmonic oscillator and orbital problems can be solved with high accuracy using the midpoint numerical method.
- N-body problems can be simulated with high precision by computers.