Motion in One Dimension

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Motion in One Dimension — Key Facts

Kinematics Quantities:

• Displacement ($s$): Change in position (vector, m)
• Velocity ($v$): Rate of change of displacement (m/s)
• Acceleration ($a$): Rate of change of velocity (m/s²)
• Average velocity: $v_{avg} = (u + v)/2$ (for constant acceleration)
• Instantaneous velocity: $v = ds/dt$, $a = dv/dt$

Equations of Motion (constant acceleration):

1. $v = u + at$
2. $s = ut + \frac{1}{2}at^2$
3. $v^2 = u^2 + 2as$
4. $s = \frac{(u+v)}{2}t$ (average velocity × time)

Where $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $s$ = displacement, $t$ = time.

Freely Falling Body: For objects dropped near Earth’s surface (ignoring air resistance): $a = g = 9.8$ m/s² downward. For upward throw, $a = -g$. Velocity at maximum height = 0.

⚡ Exam tip: Sign convention matters! If upward is positive, then $g = -9.8$ m/s². Many JAMB errors come from inconsistent sign conventions.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Motion in One Dimension — JAMB UTME Study Guide

Graphical Analysis:

• Position-time graph: Slope = velocity. If slope is positive, object moving in positive direction. Horizontal line (slope = 0) means at rest. Curved line means changing velocity (acceleration).
• Velocity-time graph: Slope = acceleration. Area under the graph = displacement. If graph is above the time axis, displacement is positive.
• Acceleration-time graph: Area under graph = change in velocity.

Example: An object moving with $u = 10$ m/s, $a = -2$ m/s². After 8 s: $v = 10 + (-2)(8) = -6$ m/s. It has reversed direction. $s = (10)(-6)/2 + \frac{1}{2}(-2)(8)^2 = 20 - 64 = -44$ m (in negative direction).

Relative Velocity: If two objects A and B move along the same line with velocities $v_A$ and $v_B$, the velocity of A relative to B is $v_{AB} = v_A - v_B$. If both move in the same direction, $v_{AB}$ is the difference. If opposite directions, $v_{AB} = v_A + v_B$.

Projectile Motion (vertical component):

• Time of flight: $T = 2u\sin\theta/g$
• Maximum height: $H = u^2\sin^2\theta/(2g)$
• Range: $R = u^2\sin 2\theta/g$
• For a body thrown upward from height $h$ with speed $u$: $t_{total} = (u/g) + \sqrt{(2h/g) + (u^2/g^2)}$

Terminal Velocity: When a falling object reaches constant velocity (drag force = weight): $mg = \rho_{fluid} C_D A v_t^2/2$. For a sphere: $v_t = \sqrt{2mg/(\rho C_D A)}$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Motion in One Dimension — Comprehensive Physics Notes

Kinematics — Full Derivation:

From the definition of acceleration: $a = \frac{dv}{dt}$. Separating variables: $dv = a \cdot dt$. Integrating from $t = 0$ to $t = t$: $\int_u^v dv = a\int_0^t dt$ gives $v - u = at$. So $v = u + at$. ✓

Displacement: $v = \frac{ds}{dt}$, so $ds = v \cdot dt = (u + at)dt$. Integrating: $s = ut + \frac{1}{2}at^2$. ✓

Eliminating $t$: From $v = u + at$, $t = (v - u)/a$. Substituting: $s = u(v-u)/a + \frac{1}{2}a(v-u)^2/a^2 = (uv - u^2)/a + (v^2 - 2uv + u^2)/(2a) = (2uv - 2u^2 + v^2 - 2uv + u^2)/(2a) = (v^2 - u^2)/(2a)$. Hence $v^2 = u^2 + 2as$. ✓

Average Velocity Derivation: For constant acceleration, $v_{avg} = \frac{s}{t} = \frac{ut + \frac{1}{2}at^2}{t} = u + \frac{1}{2}at = \frac{u + (u+at)}{2} = \frac{u+v}{2}$. This only equals $(u+v)/2$ for constant acceleration.

Free Fall Equations: A body dropped from rest ($u = 0$): $s = \frac{1}{2}gt^2$, $v = gt$, $v^2 = 2gs$. A body thrown upward from ground: $v^2 = u^2 - 2gs$ (note the minus sign because $g$ acts downward).

Numericals — JAMB Style:

Q1: A ball is thrown vertically upward with 20 m/s. How high does it go? $v^2 = u^2 - 2gh$; at max height $v = 0$. So $0 = 400 - 2(10)h$. $h = 400/20 = 20$ m.

Q2: A car accelerates from 10 m/s to 30 m/s in 8 s. Distance covered? $a = (30-10)/8 = 2.5$ m/s². $s = ut + \frac{1}{2}at^2 = 10(8) + \frac{1}{2}(2.5)(64) = 80 + 80 = 160$ m.

Q3: A body dropped from a tower falls 45 m in the last 2 s. Find height of tower. Let total time = $t$. Distance in time $t$: $s_1 = \frac{1}{2}gt^2$. Distance in time $(t-2)$: $s_2 = \frac{1}{2}g(t-2)^2$. Last 2 s distance: $s_1 - s_2 = 45$. $\frac{1}{2}(10)[t^2 - (t-2)^2] = 45$. $5[4t - 4] = 45$. $20t - 20 = 45$. $t = 3.25$ s. Height $s_1 = \frac{1}{2}(10)(3.25)^2 = 52.8$ m.

Frame of Reference: Motion is always relative to an observer. If you’re on a train moving at 20 m/s and throw a ball forward at 10 m/s relative to the train, the ball’s velocity relative to the ground is 30 m/s (if same direction) or 10 m/s (if opposite).

JAMB Common Mistakes:

1. Confusing distance with displacement (scalar vs vector)
2. Using $v = u + at$ when acceleration is not constant
3. Wrong sign for $g$ in upward motion
4. Confusing time of flight with time to maximum height (time to max = half of total time of flight for symmetric launch)
5. Using range formula $R = u^2\sin 2\theta/g$ for horizontal projection only

Dimensional Analysis: $[v] = LT^{-1}$, $[a] = LT^{-2}$, $[s] = L$. The equation $v^2 = u^2 + 2as$ is dimensionally correct. $s = ut + \frac{1}{2}at^2$ is dimensionally correct because $ut$ has dimensions $L$ and $at^2$ also has dimensions $LT^{-2} \cdot T^2 = L$.

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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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