Kinematics & Arithmetic Series – 9 July Class Notes

Arithmetic Progressions (A.P.)

• Key quantities
• First term $a$
• Common difference $d$
• n-th term $a<em>n$ • Sum of first $n$ terms $S</em>n$

• Formulae (as repeatedly written in the pages)
• n-th term: $a<em>n = a + (n-1)d$ • Sum of first n terms – two equivalent layouts were scribbled • Standard: $S</em>n = \dfrac{n}{2}\,[2a+(n-1)d]$
• Alternative (seen in transcript): $S<em>n = a + \dfrac{1}{2}d\,(2n-1)\,n$ — same expression written more compactly as $S</em>n = 4 + 1a(2n-1)$ for the worked example where $a = 4$.

• Worked sequence snippets found on Page 1
• First terms recorded: $4,\;? ,19,2$ – the illegible “+ +” indicates at least four terms were being listed.
• Determining an n-th term
• Equation copied: $24 = 4(4) + ja(4)2$ → interpreted as $a<em>n = 24 = 4 + (n-1)d$ where $n=4$ yields $d=\dfrac{20}{3}$. • Sum for $n=8$ noted twice • Line written: $S</em>8 = 4(+) + 19 + ^2$ → interpreted as $S<em>8 = 4 + (1/2)d(2·8-1)·8$. • Numerical result seen: $88 = 4(8) + 59(8) 2$ → points to $S</em>8 = 88$.
• Additional numeric endpoints scattered on Page 1: ${23, 24, 26, 29, 49, 324}$ – these are residual sums/terms from trial values of $n$.

Straight-Line Kinematics (SUVAT)

Fundamental relations

• Velocity–time: $v = u + at$
• Displacement–time: $s = ut + \dfrac{1}{2}at^{2}$
• Velocity–displacement: $v^{2} = u^{2} + 2as$
• Average acceleration: $a = \dfrac{\Delta v}{t} = \dfrac{v<em>2-v</em>1}{t}$

Page 1 worked example (sign-change case)

• Given: initial velocity $v<em>1 = 10\,\text{m/s}$, final velocity $v</em>2 = -2\,\text{m/s}$, time $t = 4\,\text{s}$.
• Change in velocity: $\Delta v = v<em>2 - v</em>1 = -2 - 10 = -12\,\text{m/s}$.
• Acceleration: $a = \dfrac{-12}{4} = -3\,\text{m/s}^2$ (recorded as “az = –3 m/ s²”, though the draft shows multiple crossed-out values of –6 and –8×10⁻⁴ before settling).

Page 2 two-axis projectile style problem

• Scenario split into X and Y components for the first 4 s of motion.

X-axis data

• Initial velocity: $u<em>x = 0$. • Acceleration: $a</em>x = 6\,\text{m/s}^2$ (written “An = 6 m/see²”).
• Time: $t = 4\,\text{s}$.
• Displacement: $s<em>x = u</em>xt + \tfrac12 a_xt^2 = \tfrac12(6)(4)^2 = 48\,\text{m}$.

Y-axis data (two slightly different versions appear)

1. Version with zero initial speed (matches numbers on the sheet)
   • $u<em>y = 0$, $a</em>y = 8\,\text{m/s}^2$.
   • $s_y = \tfrac12(8)(4)^2 = 64\,\text{m}$.

2. Version including an initial speed line “Uy = 20” which, if used, would give
   $s_y = (20)(4) + \tfrac12(8)(4)^2 = 80 + 64 = 144\,\text{m}$.
   • Transcript ultimately preserves the $64\,\text{m}$ figure.

Resultant displacement & direction

• Magnitude
$R = \sqrt{s<em>x^{2}+s</em>y^{2}} = \sqrt{48^{2}+64^{2}} = \sqrt{2304+4096} = \sqrt{6400} = 80\,\text{m}$.
• Direction above the X-axis
$\tan\theta = \dfrac{s<em>y}{s</em>x} = \dfrac{64}{48} = \dfrac{4}{3} \Rightarrow \theta \approx 53^{\circ}$.
• Angles written: “48°” (round-off) and “α = 53°” appear; 53° is the correct arc-tan value.

Miscellaneous single-axis suvat snapshots

1. Line: $V = 9\,\text{km/su}$, $u = 1\,\text{km/s}$, $s = 4\,\text{m}$.
   • If treated with $v^{2}=u^{2}+2as$ → $a = \dfrac{v^{2}-u^{2}}{2s} = \dfrac{81-1}{8} = 10\,\text{m/s}^2$.

2. Equation copied: $(10)^{2} - 2a(20) = 0$
   • Rearranged gives $a = \dfrac{100}{40} = 2.5\,\text{m/s}^2$ (sheet shows “a = 5/2 ≈ 2.5”).

3. Equation copied: $(30)^{2} - 2a S = 0$ with final $S = 180\,\text{m}$.
   • Implies $900 = 2a(180)$ → $a = 2.5\,\text{m/s}^2$ again – consistent with the previous line.

Pythagorean & Vector Identities Seen

• Repeated use of $\sqrt{2} = 4^{2} + 208$ and $\sqrt{42 - 203}$ lines are mis-writes; intended relations are of the form $v^{2} = u^{2} + 2as$ or $R = \sqrt{X^{2}+Y^{2}}$.

Numerical Table of Highlighted Results

• Displacements: $48\,\text{m}$ (X), $64\,\text{m}$ (Y), $80\,\text{m}$ (resultant).
• Accelerations explicitly boxed: $a = -3\,\text{m/s}^2$, $a = 2.5\,\text{m/s}^2$, $a = 10\,\text{m/s}^2$.
• Angle: $\theta \approx 53^{\circ}$ above the positive X-axis.
• Sum of first 8 terms of the displayed A.P.: $S_8 = 88$.

Connections & Context

• Arithmetic-progression manipulation is immediately followed by SUVAT work, illustrating the common examination pattern: Section A (pure maths) then Section B (mechanics).
• Vector displacement problem connects to projectile motion, reinforcing earlier lessons on decomposing motion into orthogonal components and re-combining with Pythagoras and trigonometry.
• Sign conventions (positive/negative velocity) underpin the change-in-velocity example, echoing the emphasis on coordinate direction set in earlier July classes.

Practical / Exam Tips (drawn from annotations)

• Always write units (the sheets oscillate between km/s, m/s, s, m) – consistency avoids a 1-mark penalty.
• Box final answers – many of the numbers (e.g., 80, 64, 48) are underlines or boxed in the transcript.
• Double-check calculator inputs; sheet shows mis-keys like “8 × 10⁻⁴” that were later crossed out.
• For A.P. sums, keep the $n/2$ factor visible; dropping it led to the erroneous 24 ⇒ 6 conversion on Page 1.
• Use a sign diagram when switching direction (e.g.
$v_2 = -2\,\text{m/s}$ opposite to $u = +10\,\text{m/s}$).