Motion in Two Dimensions

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Motion in Two Dimensions — Key Facts Position vector: $\vec{r} = x\hat{i} + y\hat{j}$; displacement $\Delta\vec{r} = \Delta x\hat{i} + \Delta y\hat{j}$ Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$; speed $v = \sqrt{v_x^2 + v_y^2}$ Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j}$ Projectile motion: range $R = \frac{u^2\sin 2\theta}{g}$; max height $H = \frac{u^2\sin^2\theta}{2g}$; time of flight $T = \frac{2u\sin\theta}{g}$ ⚡ Exam tip: For maximum range in projectile motion, launch at 45°; at 30° and 60°, range is the same

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Motion in Two Dimensions — JAMB Physics Study Guide Projectile motion from horizontal plane:

• Horizontal component: $v_x = u\cos\theta$ (constant, since $a_x = 0$)
• Vertical component: $v_y = u\sin\theta - gt$ (varies, since $a_y = -g$)
• Position at time $t$: $x = ut\cos\theta$, $y = ut\sin\theta - \frac{1}{2}gt^2$
• Velocity at time $t$: $v = \sqrt{(u\cos\theta)^2 + (u\sin\theta - gt)^2}$

Relative motion: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$; if a boat crosses a river with current, resultant velocity is vector sum of boat velocity and river current velocity.

River crossing problems: shortest time when boat aims perpendicular to bank; shortest path when boat aims at angle $\theta = \tan^{-1}(v_b/v_r)$ where $v_b$ is boat speed in still water and $v_r$ is river current speed.

Common student mistakes: forgetting that time of flight depends only on vertical motion; using wrong sign for $g$ (always take $g = 9.8 m/s^2$ or sometimes $10 m/s^2$ in JAMB); confusing horizontal range with maximum height.

Problem-solving strategy:

1. Separate horizontal and vertical components
2. Apply equations of motion to vertical component (with $g$)
3. Apply constant velocity equation to horizontal component
4. Use time $t$ as the linking variable between the two

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Motion in Two Dimensions — Comprehensive Physics Notes

Derivation of projectile motion equations: From first principles, for a projectile launched with initial velocity $u$ at angle $\theta$:

Horizontal acceleration: $a_x = 0$ $$\frac{dv_x}{dt} = 0 \Rightarrow v_x = u\cos\theta = \text{constant}$$

Vertical acceleration: $a_y = -g$ $$v_y = u\sin\theta - gt$$ $$y = ut\sin\theta - \frac{1}{2}gt^2$$

Trajectory equation: Eliminating $t$ from $x = ut\cos\theta$: $$t = \frac{x}{u\cos\theta}$$ Substituting: $$y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$$ This is the equation of a parabola, confirming projectile motion is parabolic.

Projectile motion from height $h$ (not horizontal plane): If projectile is launched from height $H$ above ground:

• Time to ground: solve $H + ut\sin\theta - \frac{1}{2}gt^2 = 0$
• Range on ground: use the appropriate $t$ value

JAMB exam patterns:

• 2019 JAMB: A ball projected horizontally from 20m height lands at distance $x$; find initial speed
• 2018 JAMB: River crossing problem with boat speed 5 m/s and river current 3 m/s
• 2017 JAMB: Projectile fired at 30° reaches same height as one fired at 60° with same speed

Key formulas summary:

Quantity	Formula
Time of flight	$T = \frac{2u\sin\theta}{g}$
Maximum height	$H = \frac{u^2\sin^2\theta}{2g}$
Range	$R = \frac{u^2\sin 2\theta}{g}$
Equation of trajectory	$y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$

Advanced concept: Motion on an inclined plane For a projectile launched up an inclined plane at angle $\alpha$ to the horizontal, range along the incline is different from horizontal range. The angle for maximum range up an incline is $\theta = 45° + \alpha/2$.

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📊 JAMB Exam Essentials

Detail	Value
Questions	180 MCQs (UTME)
Subjects	4 subjects (language + 3 for course)
Time	2 hours
Marking	+1 per correct answer
Score	400 max (used for university admission)
Registration	January – February each year

🎯 High-Yield Topics for JAMB

• Use of English (Grammar + Comprehension) — 60 marks
• Biology for Science students — 40 marks
• Chemistry (Organic + Physical) — 40 marks
• Physics (Mechanics + Optics) — 35 marks
• Mathematics (Algebra + Geometry) — 40 marks

📝 Previous Year Question Patterns

• Q: “The process of photosynthesis requires…” [2024 Biology]
• Q: “The electronic configuration of Fe is…” [2024 Chemistry]
• Q: “Find the value of x if 2x + 5 = 15…” [2024 Mathematics]

💡 Pro Tips

• Use of English carries the most weight — master grammar rules and comprehension strategies
• JAMB syllabus is your Bible — questions come directly from it. Download and use it.
• Past questions are highly predictive — repeat patterns appear every year
• For Science students, Biology and Chemistry are high-scoring if you study NCERT-level content

🔗 Official Resources

• JAMB Official
• JAMB Syllabus

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