Mathematical Physics and Problem Solving
Mathematical Physics and Problem Solving tests the mathematics used inside physics: vectors, trigonometry, differentiation, integration, logarithms, dimensional analysis, graphs and significant figures. The goal is to choose the right mathematical tool before substituting numbers.
Core rules
- Vector magnitude |A| = √(A_x² + A_y² + A_z²); dot product A·B = AB cosθ; cross product magnitude AB sinθ with direction by the right-hand rule.
- Resolve a vector into components using sine and cosine; check which angle is given. For a projectile, horizontal and vertical motions are independent.
- Differentiation gives rate of change: v = dx/dt, a = dv/dt. Integration gives accumulation: x = ∫v dt. Area under a v-t graph is displacement.
- Dimensional analysis checks equations: both sides must have the same dimensions. Significant figures in the final answer should match the least precise given data.
- Logarithms convert products to sums and solve exponential decay; use log₁₀ or ln consistently.
Worked example
Vectors A = 3i + 4j and B = 2i - j. |A| = √(9 + 16) = 5. Dot product A·B = (3)(2) + (4)(-1) = 6 - 4 = 2. Since A·B = AB cosθ, cosθ = 2/(5√5), so the angle is about 79.7°. The positive dot product confirms an acute angle.
Traps
Using degrees in a calculus formula that requires radians; dropping vector signs; rounding intermediate values too early; and writing a final answer with more significant figures than the data justify.