Math Tricks for USAPhO

Unlike AP Physics C or F=ma, USAPhO leans heavily on estimation and other mathematical tools to solve problems.

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Common integrals to know

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Small value approximations

Often, for small values, you can take an approximation that will simplify the calculations a lot.

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Taylor expansions

NOTE: If you haven’t learned about Taylor expansions, please visit the section in AP Calculus BC.

Sometimes, when you have \(a << b\), you can perform a Taylor expansion at \(x = 0\) (this is how we derive all of the small value approximations!). If you are doing oscillations/SHM calculations, you need to take up to the second derivative term, and if you are doing any other calculations you need to take up to the first derivative term. Here is an example below:

ADD EXAMPLE LATER

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Vector Algebra

Vectors show up a lot on USAPhO, so it is especially handy to know how to do vector algebra.

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Notations

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Dot product

Definition: The dot product is one way to multiply two vectors, and result in a scalar quantity. The formula is given by: \(\vec a \cdot \vec b = a_x b_x + a_y b_y\)

Properties:

1. \[|a \cdot b| = |a||b|cos\theta\]
2. \[b \cdot a = a \cdot b\]
3. \[a \cdot a = |a|^2\]
4. If \(a \perp b\), then \(a \cdot b = 0\)

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Cross product

Definition: The cross product is one way to multiply two vectors, and result in another vector, whose direction is perpendicular to the first two. You find the direction based on the right-hand rule (as described in AP Physics C). The formula is given by:

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