1 - Mathematical Foundations for Algebra-Based Physics

Quantities, Units, and Scientific Notation

Physics uses mathematics to describe measurable relationships. A strong solution connects a physical representation to an equation, calculation, and interpretation.

A physical quantity combines a number with a unit:

For example, $\Delta x=12\ \text{m}$ means that the displacement has a numerical value of $12$ when measured in meters. A number without a unit is often incomplete or ambiguous.

The SI system provides standard units. Important mechanics units include meters for length, seconds for time, and kilograms for mass. Derived units are combinations of base units:

Metric prefixes represent powers of ten. For example, $1\ \text{km}=10^3\ \text{m}$, $1\ \text{ms}=10^{-3}\ \text{s}$, and $1\ \mu\text{m}=10^{-6}\ \text{m}$. Scientific notation makes very large and small values easier to manipulate.

Takeaway: Include units in every physical answer and use standard units whenever possible.

Conversion and Dimensional Checks

A conversion factor is a ratio equal to one, so multiplying by it changes the unit without changing the physical quantity. To convert $72\ \text{km/h}$ to meters per second, arrange the factors so unwanted units cancel:

Dimensional analysis provides a second check. Dimensions identify the type of quantity independently of its particular unit. Common dimensions include length $[L]$, time $[T]$, and mass $[M]$. Thus,

In $v_f=v_i+at$, the product $at$ has dimensions $[L]/[T]$, so it can be added to $v_i$. In contrast, $v_f=v_i+a$ is invalid because velocity and acceleration have different dimensions.

Dimensional consistency is necessary but not sufficient: two physically different equations can have the same dimensions.

Takeaway: Arrange conversion factors to cancel units, then check that every equation adds and equates quantities with compatible dimensions.

Scalars, Vectors, and Components

A scalar has magnitude and unit only. Mass, time, temperature, energy, and speed are examples. A vector has magnitude and direction; displacement, velocity, acceleration, force, momentum, and torque are examples.

In one dimension, choose a positive direction. If right is positive, $+4.0\ \text{m}$ represents $4.0\ \text{m}$ to the right and $-4.0\ \text{m}$ represents $4.0\ \text{m}$ to the left. The sign communicates direction relative to the chosen axes.

A coordinate system specifies an origin, positive directions, and a scale. In two dimensions, the usual axes are $x$ and $y$. For an inclined plane, axes parallel and perpendicular to the surface can simplify the analysis. The choice of axes changes the representation, not the physics.

For a vector $\vec A$ of magnitude $A$ at angle $\theta$ above the positive $x$-axis, its Vector components are

and

Given components, reconstruct the magnitude with

The signs of the components determine the quadrant. If $\vec A=(3.0\hat{\imath}+4.0\hat{\jmath})\ \text{m}$ and $\vec B=(-1.0\hat{\imath}+2.0\hat{\jmath})\ \text{m}$, then

with magnitude

Takeaway: Choose axes deliberately, assign signs consistently, and use components to add vectors accurately.

Graphs, Slopes, and Areas

Graphs translate physical relationships into visual form. First inspect the axes, scale, units, and shape before using a graph quantitatively.

The slope is the change in the vertical quantity divided by the change in the horizontal quantity:

On a position-versus-time graph,

A steeper slope means a greater velocity magnitude, while a negative slope indicates negative velocity. On a velocity-versus-time graph,

A horizontal velocity-versus-time graph represents constant velocity and zero acceleration.

The signed area under a velocity-versus-time graph gives displacement. For constant velocity,

The signed area under an acceleration-versus-time graph gives the change in velocity:

Area below the time axis contributes negatively. A curved graph has a changing slope; the slope at one point is the instantaneous slope, found using a tangent line.

A linear relationship has the form

For constant velocity, $x=x_0+vt$ is linear in time, with slope $v$ and vertical intercept $x_0$.

Takeaway: Use slope for rates of change and signed area for accumulated changes, while always interpreting signs from the axes.

A Reliable Problem-Solving Workflow

A reliable physics solution separates representation, reasoning, and arithmetic.

1. Represent the situation. Sketch the arrangement, identify the system, choose axes, and label known and unknown quantities. Add vectors, forces, motion diagrams, tables, or graphs when useful.

2. List knowns and the target. Include units. For example, $v_i=3.0\ \text{m/s}$, $a=2.0\ \text{m/s}^2$, and $t=4.0\ \text{s}$, with target $v_f$.

3. Choose a relationship. Select an equation whose assumptions match the situation. For constant acceleration,

1. Solve symbolically first. Then substitute:

1. Check the result. Verify the units, sign, magnitude, graph or diagram, and limiting cases.

A limiting-case check asks what happens when a variable takes a simple value. If $t=0$ in $v_f=v_i+at$, the equation gives $v_f=v_i$, as expected.

Keep units throughout the calculation and cancel them as algebraic factors. Retain extra digits during intermediate steps, then apply significant figures to the final result. Estimation provides an independent plausibility check; a walking speed near $1\ \text{m/s}$ should not emerge as $10^6\ \text{m/s}$.

Takeaway: Represent first, identify the target, choose an applicable relationship, solve symbolically, calculate with units, and evaluate the result.

Proportionality, Estimation, and Precision

Proportional reasoning reveals how a quantity changes when other relevant variables remain fixed. If

then doubling $x$ doubles $y$. If

then doubling $x$ multiplies $y$ by four. If

then doubling $x$ reduces $y$ by a factor of two.

For kinetic energy,

if mass remains constant and speed changes from $v$ to $2v$, then

This reasoning helps compare scenarios and check whether a numerical answer has the correct trend.

Use estimation to establish a plausible scale before or after detailed calculation. If an answer differs drastically from the estimate, recheck unit conversions, arithmetic, equation selection, and powers of ten. Precision should reflect the information supplied rather than the number of digits displayed by a calculator.

Takeaway: Predict trends before calculating, and use proportionality, estimation, limiting cases, units, and significant figures as independent checks.