Mathematical Physics and Problem Solving
Mathematical physics and problem solving tests vectors, trigonometry, calculus basics, dimensional analysis, graphs, approximations, and multi-step numerical reasoning. The exam expects you to choose a method, keep units, and carry signs through intermediate steps.
Core rules and concepts
- Resolve vectors into components. Dot product A·B = AB cos(theta) gives work and projection; cross product magnitude AB sin(theta) gives torque and magnetic force.
- Derivatives describe rates: velocity is dx/dt, acceleration is dv/dt. Integrals accumulate: displacement is the area under a velocity-time graph.
- Dimensional analysis checks equations and converts units. Only like dimensions add.
- For small angles, sin(theta) is approximately theta in radians and cos(theta) is approximately 1 - theta^2/2.
- In graphs, slope = Delta y/Delta x and area = y times x. Log graphs turn power laws into straight lines.
- Keep significant figures consistent with the least precise input, and avoid rounding until the final step.
Worked example
Find the angle between A = 2i + 3j and B = 4i - j. Dot product = (2)(4) + (3)(-1) = 5. Magnitudes are sqrt(13) and sqrt(17). Thus cos(theta) = 5/sqrt(221) = 0.336, giving theta about 70 degrees. The positive dot product confirms an acute angle.
Traps that cost marks
- Dropping negative signs when resolving components.
- Adding quantities with different dimensions.
- Using degrees in calculus formulas that require radians.
- Reading a graph slope without converting units.
- Rounding intermediate values so the final answer drifts.