Kinematics Chapter 2: Master Revision Guide
Definitions · formulas · derivations · graphs · worked examples · common mistakes
1. Symbols and SI Units
| Symbol | Meaning | SI unit | Type |
|---|---|---|---|
| d | distance | metre (m) | scalar |
| s | displacement | metre (m) | vector |
| u | initial velocity | m/s | vector |
| v | final velocity | m/s | vector |
| a | acceleration | m/s² | vector |
| t | time | second (s) | scalar |
| g | magnitude of gravitational acceleration | 9.8 m/s² (approximately 10) | downward |
2. Quick Formula Sheet
speed = distance/time
velocity = displacement/time
a = (v − u)/t
v = u + at
s = ut + ½at²
v² = u² + 2as
v̄ = total displacement/total time
Sₙ = u + (a/2)(2n − 1)
t↑ = u/g; H = u²/(2g); T = 2u/g
v = gt; h = ½gt²; v² = 2gh
3. Definitions
Motion, rest and reference point
Motion: a body is in motion when its position changes with time relative to a chosen reference point. Rest: its position does not change relative to that reference. Motion is relative: a seated passenger is at rest relative to the bus but moving relative to the road.
Scalar and vector quantities
A scalar has magnitude only: distance, speed, time and mass. A vector has magnitude and direction: displacement, velocity and acceleration. “5 m” is a scalar distance; “5 m east” is a vector displacement.
Distance and displacement
Distance is the total path length travelled. It is scalar and never negative. Displacement is the straight-line change from initial to final position, with direction. Its magnitude is always ≤ distance.
Speed, velocity and their averages
Speed is distance covered per unit time. Velocity is displacement per unit time in a specified direction. Average speed = total distance/total time. Average velocity = total displacement/total time. Instantaneous speed or velocity is the value at one particular instant.
Uniform and non-uniform motion
Uniform velocity: equal displacements in equal time intervals in the same direction. Non-uniform velocity: unequal displacements in equal times, or a changing direction. Uniform speed in a circle is not uniform velocity because direction continuously changes.
Acceleration and retardation
Acceleration is rate of change of velocity: a = Δv/Δt. It occurs when speed changes, direction changes, or both. Retardation/deceleration means acceleration opposite to the velocity, causing speed to decrease. Negative acceleration does not automatically mean slowing; the relative directions matter.
Types of motion
- Translational: the whole body changes position; rectilinear along a straight line or curvilinear along a curve.
- Rotational: body turns about an axis, such as a wheel.
- Circular: object follows a circular path.
- Oscillatory: repeated to-and-fro motion about a mean position, such as a pendulum.
- Random: irregular path, such as a dust particle in air.
Projectile, trajectory and free fall
A projectile is an object that, after being projected, moves under gravity alone (air resistance ignored). Its path is the trajectory. An oblique projectile is launched at an angle to the horizontal. Free fall is motion under gravity alone. Near Earth, all freely falling bodies have the same acceleration g if air resistance is neglected.
4. Equations of Uniformly Accelerated Motion
Derivation 1: v = u + at
Acceleration is change in velocity per unit time:
a = (v − u)/t
at = v − u
v = u + at
Derivation 2: s = ut + ½at² (average-velocity method)
For constant acceleration, average velocity = (u + v)/2.
s = [(u + v)/2]t
Using v = u + at:
s = [(u + u + at)/2]t = [u + ½at]t
s = ut + ½at²
Graphical derivation: s = ut + ½at²
Displacement equals area under the velocity-time graph:
s = rectangle + triangle = ut + ½(v − u)t. Since v − u = at, s = ut + ½at².
Derivation 3: v² = u² + 2as
Start with s = [(u + v)/2]t and use t = (v − u)/a:
s = [(u + v)/2][(v − u)/a]
2as = (u + v)(v − u) = v² − u²
v² = u² + 2as
Distance travelled in the nth second
Total displacement after n seconds: sₙ = un + ½an². After n − 1 seconds: sₙ₋₁ = u(n − 1) + ½a(n − 1)².
Sₙ = sₙ − sₙ₋₁
Sₙ = u + (a/2)(2n − 1)
5. Motion Graphs
- s–t graph: slope gives velocity; horizontal means rest; straight slope means uniform velocity; curved line means changing velocity.
- v–t graph: slope gives acceleration; signed area gives displacement; horizontal means constant velocity.
- a–t graph: signed area gives change in velocity; line on the time axis means zero acceleration.
6. Gravity and Vertical Motion
Sign convention for g
The sign depends on the chosen positive direction, not on whether the object is moving upward or downward.
- If upward is positive: a = −g throughout.
- If downward is positive: a = +g throughout.
At maximum height v = 0, but acceleration remains g downward.
Body dropped from rest
Put u = 0 and a = g in the motion equations:
v = gt; h = ½gt²; v² = 2gh.
Vertical projection: time of ascent, maximum height and total time
Choose upward positive, so a = −g. At the top, v = 0.
From v = u − gt: t↑ = u/g.
From v² = u² − 2gH: H = u²/(2g).
If it returns to the same level, descent time equals ascent time: T = 2u/g.
Eliminating u using u = gT/2 gives H = gT²/8.
7. Average Speed and Velocity
Equal time intervals
If velocities v₁ and v₂ act for equal times, average velocity = (v₁ + v₂)/2.
Equal distances
For equal distances travelled at speeds v₁ and v₂:
Let each distance be d. Total distance = 2d; total time = d/v₁ + d/v₂.
average speed = 2v₁v₂/(v₁ + v₂)
Three equal distances
For equal distances at speeds v₁, v₂ and v₃:
average speed = 3/(1/v₁ + 1/v₂ + 1/v₃).
8. Projectile Motion Essentials
Ignoring air resistance, horizontal velocity remains constant while vertical velocity changes due to gravity. Components: uₓ = u cosθ, uᵧ = u sinθ. For launch and landing at the same level:
- time of flight: T = 2u sinθ/g
- maximum height: H = u²sin²θ/(2g)
- range: R = u²sin2θ/g
Range is maximum at 45° because sin90° = 1. Complementary angles such as 30° and 60° have equal range because sin60° = sin120°, although 60° goes higher and stays airborne longer.
9. Common Mistakes Checklist
- Do not confuse distance with displacement or speed with velocity.
- Convert units before substituting: 1 km/h = 5/18 m/s.
- Choose a positive direction and keep signs consistent.
- At maximum height, velocity is zero but acceleration is not.
- The standard three equations require constant acceleration.
- Area below a velocity-time axis is negative displacement.
- For average speed, use total distance/total time—not usually the simple arithmetic mean.
- In circular motion, constant speed can still mean changing velocity.
10. Mixed Worked Revision Example
Final velocity: v = u + at = 5 + 2(6) = 17 m/s.
Displacement: s = ut + ½at² = 5(6) + ½(2)(36) = 66 m.
Sixth-second distance: S₆ = u + (a/2)(2×6 − 1) = 5 + 1(11) = 16 m.
Check: distance in first 5 s = 5(5) + ½(2)(25) = 50 m; 66 − 50 = 16 m.
