9.1 Units, Measurements & Kinematics

9.1 Units, Measurements & Kinematics

Quick Answer: Mechanics MCQs on the PAF Aeronautical Engineering initial test reward SI fluency, dimensional checks, and clean use of the constant-acceleration equations. Physics is commonly weighted near 30% of the academic battery and sits at FSc Pre-Engineering depth—noticeably heavier than GD Pilot entry—so treat units and 1-D motion as non-negotiable foundations before forces and energy.

Physics opens the academic side of the PAF Aeronautical Engineering (CAE / engineering branch) pathway because every later topic—Newton’s laws, circular motion, fluids, thermo—rests on measurable quantities and consistent units. Selection-centre e-testing formats are not fully standardised in one public PDF for every induction cycle; coaching reports often describe roughly ~50 physics MCQs in about ~25 minutes, with separate mathematics and English papers. Always verify the current instructions on joinpaf.gov.pk and your registration slip.

Why Units Come First on an Engineering Paper

An aeronautical engineer cannot confuse mass with weight, or speed with velocity, and still reason safely about aircraft loads. The exam mirrors that discipline: many “trick” items are unit traps or dimensional mismatches, not hard algebra.

SI Base Quantities Used in Mechanics

Derived mechanical units you must recognise instantly:

Worked unit check: Convert 72 km/h to m/s.

$72\,\text{km/h} = 72 \times \frac{1000\,\text{m}}{3600\,\text{s}} = 20\,\text{m/s}$

Memorise the shortcut: divide km/h by 3.6 to get m/s (or multiply m/s by 3.6 for km/h).

Significant Figures, Precision & Error Sense

FSc papers rarely ask for formal absolute-error proofs, but they do expect you to know that a result cannot be more precise than the least precise measurement feeding it. If lengths are given to two significant figures, quoting acceleration to five decimals is incorrect exam practice. Prefer answers that match the same order of precision as the data.

Percentage error (concept): if a measured length is $L \pm \Delta L$, then

$\text{Percentage error} = \frac{\Delta L}{L} \times 100\%$

When quantities multiply, relative errors add (to first order)—useful if an MCQ asks which measurement dominates uncertainty.

Dimensional Analysis

Every physical equation must be dimensionally homogeneous: both sides share the same combination of mass (M), length (L), and time (T).

Common dimensions:

Worked example: Is $v = u + at$ dimensionally consistent?

• $[u] = LT^{-1}$
• $[at] = (LT^{-2})(T) = LT^{-1}$

Yes—both terms are velocities. Dimensional analysis cannot prove an equation is true (numerical coefficients can still be wrong), but it can eliminate distractors that mix energy with force or acceleration with velocity.

Another check: kinetic energy $\tfrac{1}{2}mv^{2}$ has dimensions $[M][LT^{-1}]^{2} = [ML^{2}T^{-2}]$, matching work. Potential energy $mgh$ has $[M][LT^{-2}][L] = [ML^{2}T^{-2}]$—same family.

Scalars vs Vectors

Distance is the path length travelled (scalar). Displacement is the straight-line change in position from start to finish (vector). A drone that flies 3 km east then 4 km north travels 7 km (distance) but has displacement magnitude 5 km (by Pythagoras).

Average vs Instantaneous Motion

$\text{Average speed} = \frac{\text{total distance}}{\text{total time}}$

$\vec{v}_{\text{avg}} = \frac{\Delta \vec{s}}{\Delta t}$

Instantaneous velocity is the limit of average velocity as the time interval shrinks toward zero—the slope of the position–time graph at a point. Instantaneous acceleration is the slope of the velocity–time graph.

Graph reading tips that appear on MCQs:

• Position–time: slope = velocity; curved graph ⇒ changing velocity ⇒ acceleration present.
• Velocity–time: slope = acceleration; area under the curve = displacement.
• Acceleration–time: area under the curve = change in velocity.

Equations of Motion (Constant Acceleration)

When acceleration $a$ is constant along a straight line:

$v = u + at$

$s = ut + \tfrac{1}{2}at^{2}$

$v^{2} = u^{2} + 2as$

Also useful: $s = \dfrac{(u+v)}{2}t$ (average velocity × time).

Sign convention: pick a positive direction and stick to it. If upward is positive, free-fall acceleration is $a = -g$.

Worked Example 1 — Runway Acceleration

An aircraft starts from rest and accelerates uniformly at $3.0,\text{m/s}^{2}$ for $12,\text{s}$. Find final speed and distance travelled.

$v = 0 + (3.0)(12) = 36\,\text{m/s}$

$s = 0 + \tfrac{1}{2}(3.0)(12)^{2} = 216\,\text{m}$

Worked Example 2 — Using $v^{2} = u^{2} + 2as$

A vehicle slows from $25,\text{m/s}$ to rest in $50,\text{m}$. Find acceleration.

$0 = (25)^{2} + 2a(50)$

$0 = 625 + 100a \implies a = -6.25\,\text{m/s}^{2}$

The negative sign means deceleration opposite the chosen positive direction of motion.

Free Fall Near Earth’s Surface

Neglecting air resistance, all objects near Earth’s surface accelerate at $g \approx 9.8,\text{m/s}^{2}$ downward (often take $g = 10,\text{m/s}^{2}$ in quick MCQs if the stem allows).

Dropped from rest: $u = 0$, $a = +g$ if downward is positive.

$v = gt, \quad s = \tfrac{1}{2}gt^{2}, \quad v^{2} = 2gs$

Thrown upward with speed $u$: time to apex $t = u/g$; maximum height $H = u^{2}/(2g)$; total flight time for return to launch height $T = 2u/g$.

Relative Velocity (1-D and Simple 2-D)

If A and B move along a line,

$\vec{v}_{AB} = \vec{v}_{A} - \vec{v}_{B}$

Two cars travelling the same direction at $30,\text{m/s}$ and $20,\text{m/s}$ have relative speed $10,\text{m/s}$. Opposite directions add magnitudes for closing speed.

Crosswind / riverboat intuition (preview for later vectors): the ground velocity is the vector sum of velocity relative to the medium and the medium’s velocity. Engineering papers love asking which heading cancels a cross component—resolve perpendicular components.

Exam Strategy for This Section

1. Convert all quantities to SI before plugging into formulas.
2. Check dimensions if two options look numerically close.
3. Draw a quick axis and mark $u$, $a$, and the unknown—most mistakes are sign errors.
4. Pace yourself: if the paper is near the commonly reported ~25-minute physics window, spend seconds—not minutes—on unit conversions you already know.

Master units and kinematics thoroughly; Section 9.2 builds Newton’s laws and energy directly on these definitions.