Problem-Solving: Mechanics

Expert problem-solvers do not start with formulas — they start by choosing a law: does this need Newton, energy, or momentum? Collisions call for momentum, smooth height changes call for energy, and "find the force" calls for Newton. Multi-stage problems chain different laws across different stages.

Mechanics problems follow a systematic approach. Applying a clear strategy prevents errors and builds physical intuition.

General strategy

1. Draw a diagram with all forces and a coordinate system.
2. List the knowns and unknowns.
3. Choose the right principle:

1. Write the equations and solve algebraically before substituting numbers.
2. Check units, limiting cases, and sign conventions.

Dimensional analysis — Every equation must be dimensionally consistent:

$$[F] = \text{kg}\cdot\text{m/s}^2, \qquad [E] = \text{kg}\cdot\text{m}^2/\text{s}^2$$

g = 9.8      # m/s^{2}
m = 2.0      # kg
v0 = 15.0    # m/s
theta = 45   # degrees
import math
R = v0**2 * math.sin(2*math.radians(theta)) / g
print(f"Range = {R:.2f} m")

Common pitfalls

• Forgetting that $\vec{F}_{\text{net}}$ is a vector sum, not a scalar sum.
• Using energy conservation when non-conservative forces do work.
• Confusing $g = 9.8$ with $g = -9.8$ — the sign depends on your axis choice.

Tip: When stuck, count equations and unknowns. If you have fewer equations than unknowns, you need another physical principle or constraint.

Common pitfall: Applying energy conservation across a sticking collision. Perfectly inelastic impacts always destroy kinetic energy; only momentum bridges that moment. Split the problem at the collision and switch tools.