BSc Physics 1st Year (TU) Unit 1: Kinematics — Complete Notes, Derivations, Numericals & Important Questions
Introduction
Kinematics is the branch of mechanics that describes the motion of objects — how they move through space and time — without asking why they move. The "why" is dynamics (Newton's Laws). Kinematics is the "what" and "how": the position, displacement, velocity, acceleration, and the mathematical relationships that connect them.
It seems deceptively simple at first glance. Three equations of motion. A few definitions. Some graphs. But kinematics is actually the foundation on which all of classical mechanics rests. Every problem in dynamics, projectile motion, circular motion, rotational mechanics, and orbital mechanics begins by setting up a kinematic description. Students who understand kinematics properly — not just the formulas but what each variable physically means — solve every subsequent mechanics problem more cleanly.
For TU exams, kinematics delivers reliable marks. The derivations of the three equations of motion appear regularly. Projectile motion is a standard long question. Graphical analysis and relative velocity problems appear as either short or numerical questions. This guide covers all of it: the theory, the full derivations, five worked numericals, a complete formula sheet, and the past-question patterns.
Learning Objectives
After studying this unit you should be able to:
- Distinguish between scalars and vectors, and between distance/displacement and speed/velocity
- Define average and instantaneous velocity and acceleration
- Derive the three equations of uniform motion from first principles
- Analyze motion using displacement-time, velocity-time, and acceleration-time graphs
- Solve projectile motion problems (maximum height, range, time of flight)
- Calculate relative velocity between two moving objects
1. Rest and Motion
An object is said to be at rest when its position does not change with time relative to a chosen reference point. An object is said to be in motion when its position changes with time relative to that reference point.
Both rest and motion are relative — an object may be at rest relative to one observer and in motion relative to another. A passenger sitting in a moving train is at rest relative to other passengers but in motion relative to someone standing on the platform. This is why we always define a reference frame — a coordinate system attached to a specific observer — before describing any motion.
2. Types of Motion
Translational Motion: Every point of the object moves the same distance in the same direction in the same time. A car moving in a straight line is translational.
- Rectilinear (linear): Motion along a straight line. A falling ball, a car on a straight road.
- Curvilinear: Motion along a curved path. A car going around a bend.
Rotational Motion: The object rotates about an axis. A spinning top, a wheel on an axle.
Circular Motion: A special case of rotational motion where every point on the object moves in a circle. A satellite orbiting Earth, a stone on a string.
Oscillatory (Vibratory) Motion: The object moves back and forth about a mean position. A pendulum, a guitar string, a spring.
3. Scalars and Vectors
| Scalar | Vector |
|---|---|
| Has magnitude only | Has both magnitude and direction |
| Added by simple arithmetic | Added using vector rules |
| Distance, speed, mass, time, energy, temperature | Displacement, velocity, acceleration, force, momentum |
| Example: 5 km | Example: 5 km North |
Understanding this distinction is critical: speed and velocity are different quantities. So are distance and displacement.
4. Position, Distance, and Displacement
Position: Location of an object relative to a reference point, described by a coordinate.
Distance: The total path length travelled by an object, regardless of direction. Scalar.
Displacement: The straight-line distance from the initial position to the final position, along with direction. Vector.
| Aspect | Distance | Displacement |
|---|---|---|
| Type | Scalar | Vector |
| Value | Always positive | Can be positive, negative, or zero |
| Path dependence | Depends on path taken | Independent of path |
| Example | You walk 4 km East then 3 km West: distance = 7 km | displacement = 1 km East |
5. Speed and Velocity
Speed = distance / time. Scalar.
Velocity = displacement / time. Vector. Velocity has the same direction as displacement.
Average Speed = Total distance / Total time
Average Velocity = Total displacement / Total time = Δx / Δt = (x₂ − x₁) / (t₂ − t₁)
Instantaneous Velocity: Velocity at a specific instant in time.
v = lim(Δt→0) Δx/Δt = dx/dt
An object can have constant speed but changing velocity — for example, a car moving at constant speed around a curve is always changing direction, so its velocity vector is always changing.
6. Acceleration
Acceleration is the rate of change of velocity with time.
Average Acceleration = Δv / Δt = (v − u) / t
where u = initial velocity, v = final velocity, t = time interval.
Instantaneous Acceleration = dv/dt = d²x/dt²
Types:
- Positive acceleration: Velocity increasing in the positive direction (speeding up)
- Negative acceleration (deceleration): Velocity decreasing (slowing down)
- Zero acceleration: Constant velocity (no change)
SI Unit of acceleration: m/s²
7. Equations of Uniformly Accelerated Motion — Derivations
These three equations apply when acceleration is constant. They are the most commonly tested derivations in TU kinematics.
Symbols:
- u = initial velocity (m/s)
- v = final velocity (m/s)
- a = constant acceleration (m/s²)
- t = time (s)
- s = displacement (m)
First Equation: v = u + at
Derivation from definition of acceleration:
By definition: a = (v − u) / t
Rearranging: v − u = at
∴ v = u + at ✓
This equation relates velocity, acceleration, and time. No displacement involved.
Second Equation: s = ut + ½at²
Derivation:
Average velocity = (u + v) / 2 (for uniform acceleration)
Displacement = average velocity × time
s = [(u + v) / 2] × t
Substitute v = u + at:
s = [(u + u + at) / 2] × t = [(2u + at) / 2] × t
s = ut + ½at²
∴ s = ut + ½at² ✓
This equation relates displacement, initial velocity, acceleration, and time. No final velocity involved.
Third Equation: v² = u² + 2as
Derivation:
From the first equation: v = u + at → t = (v − u) / a
Substitute into the second equation: s = ut + ½at²
s = u·[(v−u)/a] + ½a·[(v−u)/a]²
s = u(v−u)/a + ½(v−u)²/a
2as = 2u(v−u) + (v−u)²
2as = 2uv − 2u² + v² − 2uv + u²
2as = v² − u²
∴ v² = u² + 2as ✓
This equation relates velocity, displacement, and acceleration. No time involved.
8. Graphical Analysis of Motion
Displacement-Time (x-t) Graph
The slope of the x-t graph at any point = instantaneous velocity (dx/dt).
| Shape | What it means |
|---|---|
| Horizontal line | Object at rest (v = 0) |
| Straight line with positive slope | Constant positive velocity |
| Straight line with negative slope | Constant negative velocity |
| Upward-curving line | Increasing velocity (positive acceleration) |
| Downward-curving line | Decreasing velocity (decelerating) |
Velocity-Time (v-t) Graph
The slope of the v-t graph = acceleration (dv/dt). The area under the v-t graph = displacement.
| Shape | What it means |
|---|---|
| Horizontal line | Constant velocity (a = 0) |
| Straight line with positive slope | Constant positive acceleration |
| Straight line with negative slope | Constant deceleration |
| Area above x-axis | Positive displacement |
| Area below x-axis | Negative displacement |
Acceleration-Time (a-t) Graph
The area under the a-t graph = change in velocity (Δv).
For uniform acceleration: a horizontal line at constant height.
9. Projectile Motion
Projectile motion occurs when an object is launched with an initial velocity and moves under gravity alone (ignoring air resistance). The key insight: horizontal and vertical motions are independent.
Setup: Object launched at angle θ with initial speed u.
- Horizontal component of initial velocity: uₓ = u cos θ
- Vertical component of initial velocity: uᵧ = u sin θ
- Horizontal acceleration: aₓ = 0 (no horizontal force)
- Vertical acceleration: aᵧ = −g (gravity downward)
Key Formulas for Projectile Motion
Horizontal position: x = u cos θ · t
Vertical position: y = u sin θ · t − ½gt²
Velocity at time t:
- Horizontal: vₓ = u cos θ (constant throughout)
- Vertical: vᵧ = u sin θ − gt
Time to reach maximum height (when vᵧ = 0): t_max = u sin θ / g
Maximum height: H = (u sin θ)² / (2g) = u²sin²θ / 2g
Total time of flight (returns to same level): T = 2u sin θ / g
Horizontal range: R = u² sin 2θ / g
Maximum range occurs at θ = 45°: R_max = u² / g
The trajectory equation (eliminating t) gives a parabola: y = x tan θ − gx² / (2u²cos²θ)
10. Relative Velocity
When two objects are both moving, the velocity of one object as seen by an observer on the other is called relative velocity.
Relative velocity of A with respect to B: v_AB = v_A − v_B
Cases:
- Same direction: v_AB = v_A − v_B (relative velocity is the difference)
- Opposite directions: v_AB = v_A + v_B (relative velocity is the sum)
- At angle θ to each other: |v_AB| = √(v_A² + v_B² − 2v_A v_B cosθ)
Example: A car A moves at 60 km/h and car B moves at 40 km/h in the same direction. Relative velocity of A with respect to B = 60 − 40 = 20 km/h in the direction of motion.
11. Solved Numerical Problems
Problem 1 — Basic Equations of Motion
A car starts from rest and accelerates uniformly at 3 m/s². Find: (a) velocity after 5 seconds, (b) distance covered in 5 seconds.
Given: u = 0, a = 3 m/s², t = 5 s
(a) v = u + at = 0 + 3 × 5 = 15 m/s
(b) s = ut + ½at² = 0 + ½ × 3 × 25 = 37.5 m
Problem 2 — Find Acceleration and Distance
A train moving at 72 km/h is brought to rest in 10 seconds. Find: (a) deceleration, (b) distance covered during braking.
Given: u = 72 km/h = 20 m/s, v = 0, t = 10 s
(a) a = (v − u) / t = (0 − 20) / 10 = −2 m/s² (deceleration)
(b) s = (u + v)/2 × t = (20 + 0)/2 × 10 = 100 m
Problem 3 — Third Equation of Motion
A ball is thrown upward with initial velocity 20 m/s. Find: (a) maximum height reached, (b) time to reach maximum height. (g = 10 m/s²)
At maximum height, v = 0. Taking upward as positive: a = −10 m/s²
(a) v² = u² + 2as → 0 = 400 + 2(−10)s → s = 400/20 = 20 m
(b) v = u + at → 0 = 20 − 10t → t = 2 s
Problem 4 — Projectile Motion
A ball is kicked at 30 m/s at an angle of 30° to the horizontal. Find: (a) maximum height, (b) range, (c) time of flight. (g = 10 m/s²)
Given: u = 30 m/s, θ = 30°, sin30° = 0.5, cos30° = 0.866
(a) H = u²sin²θ / 2g = (900 × 0.25) / 20 = 225/20 = 11.25 m
(b) R = u²sin2θ / g = 900 × sin60° / 10 = 900 × 0.866 / 10 = 77.9 m
(c) T = 2u sinθ / g = 2 × 30 × 0.5 / 10 = 3 s
Problem 5 — Relative Velocity
Two trains approach each other on parallel tracks. Train A moves at 80 km/h and Train B moves at 60 km/h. Find the relative velocity of A with respect to B.
Moving towards each other means opposite directions. Taking A's direction as positive: v_A = +80 km/h, v_B = −60 km/h
v_AB = v_A − v_B = 80 − (−60) = 140 km/h
The trains approach each other at 140 km/h.
12. Formula Quick Sheet
| Formula | What it gives |
|---|---|
| v = u + at | Velocity at time t |
| s = ut + ½at² | Displacement at time t |
| v² = u² + 2as | Velocity after displacement s |
| v_avg = s/t | Average velocity |
| a = (v−u)/t | Average acceleration |
| H = u²sin²θ / 2g | Maximum height (projectile) |
| T = 2u sinθ / g | Time of flight (projectile) |
| R = u²sin2θ / g | Range (projectile) |
| R_max = u²/g | Maximum range (at θ = 45°) |
| v_AB = v_A − v_B | Relative velocity |
13. Important Exam Questions
Long Questions (Often 10 Marks)
- Derive the three equations of uniformly accelerated motion from first principles.
- Explain projectile motion. Derive expressions for maximum height, time of flight, and horizontal range.
- Explain the graphical analysis of motion using displacement-time and velocity-time graphs.
- Derive the expression for relative velocity of two objects moving at an angle to each other.
- Explain translational and rotational motion with examples.
Short Questions (5 Marks)
- Define displacement. How does it differ from distance?
- What is instantaneous velocity?
- Define acceleration. What is the unit of acceleration?
- Differentiate between scalars and vectors with examples.
- State the conditions under which the three equations of motion are valid.
- At what angle is the horizontal range of a projectile maximum? Why?
- What is relative velocity? Give one example.
Numerical Questions
- A car decelerates from 25 m/s to rest in 50 m. Find the deceleration and time taken.
- A projectile is fired at 45° with initial speed 40 m/s. Find the range and maximum height.
- Two cars move in the same direction at 90 km/h and 60 km/h. Find their relative velocity.
14. MCQs
1. Displacement is: A. Total path length B. A scalar quantity C. Always positive D. A vector quantity ✅
2. The slope of a velocity-time graph gives: A. Displacement B. Acceleration ✅ C. Speed D. Distance
3. At maximum height, the vertical velocity of a projectile is: A. Zero ✅ B. Maximum C. Equal to initial speed D. Negative
4. For maximum range, the angle of projection should be: A. 45° ✅ B. 30° C. 60° D. 90°
5. The area under a v-t graph gives: A. Acceleration B. Displacement ✅ C. Speed D. Force
6. Two bodies moving in opposite directions have relative velocity equal to: A. Difference of their speeds B. Sum of their speeds ✅ C. Product D. Zero
15. Exam Tips
For derivation questions: Present derivations in clear numbered steps. State what you start from (definition of acceleration, for example), write every intermediate algebraic step, and box the final result. Steps earn marks; skipping steps loses them even if the answer is correct.
For numerical questions: Write the given information first (u, v, a, t, s), state which formula you'll use, substitute with units, and calculate. Always include the unit in the final answer — a number without a unit is usually marked wrong.
For projectile motion: Always resolve initial velocity into horizontal and vertical components before doing anything else. Label them clearly and treat horizontal and vertical as completely separate calculations throughout.
Draw graphs: Displacement-time and velocity-time graph questions that come with a diagram request need clearly labelled axes, correct shape of curve, and annotations explaining what each section shows.
Use SI units throughout: Convert km/h to m/s (divide by 3.6) before substituting into equations. Mixing unit systems produces wrong numerical answers.
Recommended Reference Books
- Resnick, Halliday & Walker — Fundamentals of Physics (9th Edition, Wiley)
- Sears, Zemansky, Young & Freedman — University Physics (14th Edition, Pearson)
- TU BSc Physics Reference Material (departmental notes)
Conclusion
Kinematics is where physics education begins its serious mathematical work. The three equations of motion, the decomposition of projectile motion into independent components, and the graphical interpretation of motion are tools that reappear throughout every subsequent mechanics topic. Students who can derive v = u + at, s = ut + ½at², and v² = u² + 2as from first principles — rather than just memorizing them — understand mechanics at the level TU exams reward.
Practice derivations until they're automatic, and practice numericals until you can identify which equation to use from the variables given, not from recognizing a problem "type."
For related guides: BSc CSIT Physics (PHY118) Complete Guide, Newton's Laws of Motion — Complete Guide, and Mathematics I (MTH117) Complete Guide.
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