PHY131H1/Unit 1

Unit 1 · Introduction to Physics I

Units, Vectors & Estimation

The unglamorous unit that pays for itself all term. Dimensional analysis catches wrong answers in seconds, vector components turn geometry into arithmetic, and an order-of-magnitude estimate tells you whether the calculator is lying to you.

By the end you can

• Check any equation for dimensional consistency.
• Resolve a vector into components and add vectors correctly.
• Carry significant figures through a calculation.
• Estimate an answer to an order of magnitude before computing it.

Why it matters & where it comes from

Why it exists

Physics equations relate quantities, not numbers. A length can never equal a time, so an equation that says so is wrong before you check the algebra. That single observation is the fastest error detector in the course.

In the real world

NASA lost the 125-million-dollar Mars Climate Orbiter in 1999 because one team supplied thrust in pound-seconds and the other read it as newton-seconds. The arithmetic was flawless; the units were not.

Story & open questions

The SI base units were redefined in 2019 so that all of them derive from fixed constants of nature rather than physical objects. The last artefact, the platinum-iridium kilogram in Paris, had been measurably drifting for a century.

Dimensional analysis

Definition

Every mechanical quantity is built from three dimensions: length $L$, mass $M$ and time $T$. Speed is $L/T$, acceleration is $L/T^2$, force is $ML/T^2$, energy is $ML^2/T^2$.

Rule Both sides must match

An equation is only possible if every term has the same dimensions. Terms that are added must match; a mismatch proves the equation wrong.

Example Catching an error instantly

Is $v^2 = 2ax$ dimensionally possible? Left side is $(L/T)^2 = L^2/T^2$. Right side is $(L/T^2)(L) = L^2/T^2$. They match, so the equation survives the test.

Is $v = 2ax$? Left is $L/T$, right is $L^2/T^2$. No match, so this is wrong — established without knowing any physics.

Dimensional analysis cannot catch a missing factor of $2$, since pure numbers have no dimensions. It rules equations out, never in.

Example Deriving a form

Guess the period of a pendulum from its length $L$, mass $m$ and $g$.

Try $T = L^a m^b g^c$. Dimensions: $T = L^a M^b (L/T^2)^c$.

Matching $M$: $b = 0$, so the mass cannot appear. Matching $T$: $1 = -2c$, so $c = -\tfrac12$. Matching $L$: $0 = a + c$, so $a = \tfrac12$.

So $T \propto \sqrt{L/g}$ — the correct form, from dimensions alone. Only the factor $2\pi$ is missing, and Unit 9 supplies it.

Vectors

Definition

A scalar has size only: mass, time, energy, speed. A vector has size and direction: displacement, velocity, acceleration, force. Vectors are written $\vec{A}$ and their magnitude $|\vec{A}|$ or $A$.

Rule Add by components, never by magnitudes

Resolve each vector onto axes, add the components separately, then recombine:

$A_x = A\cos\theta$ and $A_y = A\sin\theta$, with $\theta$ measured from the $+x$ axis.

$R_x = A_x + B_x$ and $R_y = A_y + B_y$.

$R = \sqrt{R_x^2 + R_y^2}$ and $\theta = \arctan(R_y / R_x)$.

Magnitudes do not add

Two forces of $3$ N and $4$ N do not make $7$ N unless they point the same way. At right angles the result is $5$ N; in opposition it is $1$ N. Only components add.

Example A full addition

Add $\vec{A}$ of magnitude $5$ at $37$ degrees to $\vec{B}$ of magnitude $3$ at $180$ degrees.

$A_x = 5\cos 37 = 4.0$, $A_y = 5\sin 37 = 3.0$.

$B_x = 3\cos 180 = -3.0$, $B_y = 0$.

$R_x = 1.0$, $R_y = 3.0$, so $R = \sqrt{1 + 9} = 3.2$ at $\arctan(3.0/1.0) = 72$ degrees.

The arctan quadrant trap

A calculator's $\arctan$ returns angles between $-90$ and $+90$ degrees, so it cannot distinguish the second quadrant from the fourth. Always sketch the resultant and check the answer lies in the quadrant the components imply, adding $180$ degrees if not.

Significant figures and estimation

Rule Precision does not improve

Multiplying or dividing: the result keeps as many significant figures as the least precise input.

Adding or subtracting: the result keeps as many decimal places as the least precise input.

Round only at the end. Rounding intermediate values accumulates error, exactly as in floating-point arithmetic.

Example Reporting sensibly

A distance of $12.5$ m measured in $2.1$ s gives $12.5 / 2.1 = 5.952380...$ on the calculator.

Both inputs have two or three significant figures, so report $6.0$ m/s. Writing $5.95$ claims a precision the measurement never had.

Example Order-of-magnitude estimate

Roughly how many breaths does a person take in a lifetime?

About $15$ breaths per minute, so about $10^4$ per day — $15 \times 60 \times 24$ is around $20{,}000$.

Times $365$ days is about $10^7$ per year; times $80$ years is about $10^9$.

A billion breaths. The point is not the exact figure but that the answer is nearer $10^9$ than $10^6$ or $10^{12}$ — enough to catch a calculator slip of three orders of magnitude.

Common pitfalls

• Mixing units in one equation. Convert everything to SI first: metres, kilograms, seconds.
• Adding vector magnitudes instead of components.
• Trusting $\arctan$ without checking the quadrant.
• Measuring the angle from the wrong axis. $\cos$ goes with the axis the angle is measured from; if the angle is from $y$, the roles swap.
• Reporting ten digits from two-digit data.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Dimensional check

Is $x = \tfrac12 a t^2$ dimensionally consistent?

Answer

Yes. Right side is $(L/T^2)(T^2) = L$, matching the left. The $\tfrac12$ is dimensionless and invisible to this test.

Practice 2 · Components

Resolve a force of $20$ N at $60$ degrees above horizontal.

Answer

$F_x = 20\cos 60 = 10$ N, $F_y = 20\sin 60 = 17.3$ N.

Practice 3 · Vector sum

Add $6$ N east and $8$ N north.

Answer

$\sqrt{36 + 64} = 10$ N at $\arctan(8/6) = 53$ degrees north of east.

Practice 4 · Opposing vectors

What is the range of possible magnitudes when adding a $5$ N and a $3$ N force?

Answer

From $2$ N (opposed) to $8$ N (aligned), any value between depending on the angle. At right angles it is $\sqrt{34} = 5.8$ N.

Practice 5 · Derive a form

Use dimensions to find how orbital speed depends on $G$, $M$ and $r$.

Solution

$G$ has dimensions $L^3/(MT^2)$. Trying $v = G^a M^b r^c$ and matching gives $a = b = \tfrac12$ and $c = -\tfrac12$, so $v \propto \sqrt{GM/r}$ — correct, with the numerical factor being exactly $1$ here.

Practice 6 · Significant figures

A car travels $150$ m in $8.2$ s. Report the speed.

Answer

$18$ m/s. The calculator says $18.29$, but $8.2$ carries only two significant figures.

Practice 7 · Estimate

Roughly how many litres of water does Toronto use in a day?

Solution

About $3$ million people at roughly $200$ litres each is about $6 \times 10^8$ litres. Any answer within a factor of a few is a success; the goal is the exponent.

Practice 8 · Quadrant

A vector has $R_x = -2$ and $R_y = 2$. Find its direction.

Solution

$\arctan(2/-2)$ gives $-45$ degrees on a calculator, but the components put the vector in the second quadrant, so the true direction is $-45 + 180 = 135$ degrees. Sketching it first would have made this obvious.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY131H1 topic map. It should cover the following items before you move on:

• SI units. Base quantities and converting to them before computing.
• Dimensional analysis. Checking consistency and deriving proportionalities.
• Vectors. Components, addition, magnitude and direction, quadrant care.
• Precision. Significant figures and order-of-magnitude estimation.

Textbook/source alignment

The calendar's topic list begins with classical kinematics, which assumes this material. Most introductory texts cover it in an opening chapter; check the syllabus on Quercus for the assigned reading.

Core idea map

Quantities carry dimensions, so equations can be checked before they are solved. Directional quantities need components, so geometry becomes arithmetic. Both tools are used in every unit that follows.

Study routine. Check units on every final answer, without exception. It costs ten seconds and catches a substantial fraction of mistakes before they reach a marker.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · What dimensions prove

Can dimensional analysis confirm an equation is right?

Guide

No. It can only rule equations out. Dimensionless factors like $\tfrac12$ or $2\pi$ are invisible to it.

Extension 2 · Scalar or vector

Is speed a vector?

Guide

No — speed is the magnitude of velocity, so it is a scalar. Velocity carries direction.

Extension 3 · Component rule

When is $A_x = A\cos\theta$ correct?

Guide

When $\theta$ is measured from the $x$ axis. Measured from $y$, the cosine and sine swap.

Extension 4 · Why estimate

What does an order-of-magnitude estimate buy you?

Guide

A check on the exponent. It catches unit-conversion and calculator errors that produce answers wrong by factors of a thousand.

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