Every definition, law, formula and worked numerical for the Sindh Board (BIEK) paper — written for Class XI. Read it straight through, or open the interactive lecture to watch each concept animate.
1 — Rest, motion & kinematics
Mechanics is the branch of physics that studies motion. It has two parts: kinematics describes how a body moves — its distance, speed and acceleration — while dynamics explains why motion changes, in terms of forces. This chapter is pure kinematics.
- Rest — a body is said to be at rest if it does not change its position with respect to its surroundings.
- Motion — a body is said to be in motion if it changes its position with respect to its surroundings.
- Frame of reference — a coordinate system (the surroundings) with respect to which position and motion are measured.
Motion is relative
Rest and motion are always relative to a chosen frame. A passenger seated in a moving bus is at rest relative to the bus and other passengers, but is in motion relative to the road and trees outside.
Types of motion
| Type | Description & example |
|---|
| Translatory | every part moves the same distance — a car on a straight road |
| Rotatory | a body spins about a fixed axis — a turning fan |
| Vibratory | to-and-fro about a mean position — a swinging pendulum |
2 — Scalars & vectors
- Scalar quantity — has magnitude only. Examples: distance, speed, time, mass, energy, temperature.
- Vector quantity — has both magnitude and direction. Examples: displacement, velocity, acceleration, force, momentum.
A vector is represented by an arrow: the length of the arrow (to a scale) is the magnitude, and the arrowhead shows the direction. So "60 km/h" is a speed (scalar), while "60 km/h due north" is a velocity (vector).
Exam point: the same numerical size can be a scalar or a vector depending on whether a direction is attached.
3 — Distance & displacement
- Distance — the total length of the path actually travelled by a body. It is a scalar and is always positive. SI unit: metre (m).
- Displacement — the shortest (straight-line) distance from the initial to the final position, directed from start to finish. It is a vector. SI unit: metre (m).
Relationshipdistance ≥ | displacement | (always)
Worth remembering: if a body moves around a circular track and returns to its starting point, the distance equals the circumference but the displacement is zero.
4 — Speed & velocity
- Speed — the rate of change of distance. A scalar. SI unit: metre per second (m/s).
- Velocity — the rate of change of displacement (i.e. speed in a given direction). A vector. SI unit: m/s.
Formulasaverage speed = total distance / total time
velocity = displacement / time
Uniform & variable velocity
- Uniform velocity — equal displacements in equal intervals of time, in a fixed direction.
- Variable velocity — velocity changes in magnitude, direction, or both.
average speed
A car covers 240 m in 12 s. Find its average speed.
v = distance / time = 240 / 12 = 20 m/s
5 — Acceleration
- Acceleration — the rate of change of velocity. A vector. SI unit: metre per second squared (m/s²).
Accelerationa = (v − u) / t
u = initial velocity · v = final velocity · t = time taken
- Uniform acceleration — velocity changes by equal amounts in equal time intervals.
- Retardation (deceleration) — negative acceleration; the body is slowing down.
find a
A bus speeds up from 10 m/s to 30 m/s in 5 s.
a = (v − u)/t = (30 − 10)/5 = 4 m/s²
6 — Distance–time graphs
A graph of distance (y-axis) against time (x-axis). The slope (gradient) of the graph gives the speed of the body.
| Shape of graph | What it shows |
|---|
| Horizontal straight line | body at rest (speed = 0) |
| Straight sloping line | uniform (constant) speed |
| Curve bending upward (steeper) | speed increasing — acceleration |
| Curve bending over (flatter) | speed decreasing — retardation |
7 — Velocity–time graphs
A graph of velocity (y-axis) against time (x-axis). It gives two important readings:
Two key resultsslope of the line = acceleration (a)
area under the line = distance / displacement (S)
| Shape | Meaning |
|---|
| Horizontal line | uniform velocity (a = 0) |
| Straight line sloping up | uniform acceleration |
| Straight line sloping down | uniform retardation |
area = distance
A body accelerates from rest to 20 m/s in 10 s.
distance = area of triangle = ½ × base × height
= ½ × 10 × 20 = 100 m
8 — Equations of motion
For a body moving in a straight line with uniform acceleration, the following three equations relate the five quantities: initial velocity u, final velocity v, acceleration a, time t and distance S.
The three equations of motion① v = u + a t
② S = u t + ½ a t²
③ 2 a S = v² − u²
Strategy: write down which of u, v, a, t, S you are given and which one you need, then choose the equation that contains exactly those letters.
use equation ②
A car starts from rest and accelerates at 2 m/s² for 6 s. Find the distance.
u = 0, a = 2, t = 6
S = ut + ½at² = 0 + ½(2)(6²) = 36 m
9 — Motion under gravity (free fall)
When a body falls freely under gravity (air resistance neglected), it moves with a constant downward acceleration g ≈ 9.8 m/s² (often taken as 10 m/s² for quick calculations). Remarkably, this is the same for all masses. The equations of motion apply with a replaced by g and S by the height h.
Equations for free fallv = u + g t
h = u t + ½ g t²
2 g h = v² − u²
dropped from rest
A stone is dropped (u = 0) and falls for 3 s. Take g = 10 m/s².
v = u + gt = 0 + 10(3) = 30 m/s
h = ½ g t² = ½(10)(3²) = 45 m
10 — Projectile motion
- Projectile — a body given an initial velocity and then allowed to move freely under gravity alone. Its path (trajectory) is a parabola.
The key idea is that a projectile has two independent motions: a horizontal motion at constant velocity (no horizontal force) and a vertical motion with acceleration g downward.
Launched with speed u at angle θHorizontal velocity = u cos θ (constant)
Time of flight T = 2u sin θ / g
Maximum height H = u² sin²θ / 2g
Range R = u² sin 2θ / g
Maximum range occurs at a launch angle of θ = 45°, because sin 2θ is largest (= 1) there.
11 — Worked numericals
final velocity (braking)
A train moving at 15 m/s brakes at 3 m/s². Find its speed after 4 s.
u = 15, a = −3, t = 4
v = u + at = 15 + (−3)(4) = 3 m/s
find the acceleration
A bike speeds up from 10 m/s to 20 m/s over 75 m. Find a.
2aS = v² − u² → a = (v² − u²)/(2S)
= (20² − 10²)/(2 × 75) = 300/150 = 2 m/s²
height of a tower
A ball is thrown up at 20 m/s. How high does it rise? (g = 10)
at the top v = 0; 2gh = v² − u² (taking up as +, a = −g)
h = u²/(2g) = 400/20 = 20 m
12 — Exam recap
- Define rest, motion and frame of reference; motion is relative.
- Scalars vs vectors with examples.
- Distance vs displacement; speed vs velocity (with units).
- Acceleration a = (v − u)/t; retardation.
- Read slope and area off motion graphs.
- Three equations of motion; free fall with g; projectile range, height and time of flight.