Board exam focus — Motion (CBSE & HBSE)

CBSE emphasises graphical interpretation (distance-time and velocity-time graphs), derivation of the three equations of motion, and uniform circular motion as a numerical and conceptual tool. HBSE (Haryana Board, NCERT-aligned) keeps the same content but weights direct numerical substitution and definition-based questions slightly more, often asking distance vs displacement distinctions and unit conversions.

Distance, Displacement, Speed, Velocity and Acceleration

Describing Motion

An object is said to be in motion if it changes its position with respect to a reference point (origin) as time passes. Motion is always relative to a chosen frame of reference.

Distance vs Displacement

Trap: If an object returns to its starting point, displacement = 0 but distance covered is NOT zero. Displacement magnitude is always less than or equal to distance.

Speed and Velocity

• Speed = distance / time. It is a scalar. SI unit: m/s (also m s⁻¹).
• Average speed = total distance / total time.
• Velocity = displacement / time. It is a vector (speed in a given direction).
• Average velocity for uniform acceleration = (u + v) / 2, where u = initial and v = final velocity.

Unit conversion: 1 km/h = 1000 m / 3600 s = 5/18 m/s. To convert km/h to m/s multiply by 5/18; m/s to km/h multiply by 18/5.

Acceleration

Acceleration is the rate of change of velocity with time.

$$a = \frac{v - u}{t}$$

• SI unit: m/s² (m s⁻²). It is a vector.
• Positive (accelerated) motion: velocity increases (a is along motion).
• Negative acceleration (retardation/deceleration): velocity decreases (a opposes motion).
• Uniform acceleration: velocity changes by equal amounts in equal time intervals (e.g. free fall).

Tip: Always assign a direction sign first. If you take the initial direction of motion as positive, a retarding force gives a negative value of a.

Uniform & Non-uniform Motion and Motion Graphs

Uniform vs Non-uniform Motion

Distance–Time (s–t) Graphs

• The slope of a distance–time graph gives speed.
• Uniform motion: straight inclined line (constant slope). Steeper line ⇒ greater speed.
• Object at rest: straight horizontal line (zero slope).
• Non-uniform motion: a curved line (slope changes). A curve bending upward shows increasing speed (acceleration).

Velocity–Time (v–t) Graphs

• The slope of a velocity–time graph gives acceleration.
• The area under a velocity–time graph gives the distance (displacement) travelled.
• Uniform velocity: horizontal straight line; area under it is a rectangle, distance = velocity × time.
• Uniform acceleration from rest: straight line rising from the origin; area is a triangle, distance = ½ × base × height = ½ × t × v.
• Uniform acceleration with initial velocity u: straight line starting at u; area is a trapezium = ut + ½(v − u)t.

Trap: Students confuse the two graphs. Remember: SLOPE of distance–time = speed; SLOPE of velocity–time = acceleration; AREA under velocity–time = distance. A negative slope on a v–t graph means retardation.

Reading a curved v–t graph

If the velocity–time graph is a curve, acceleration is non-uniform; the distance is still the area under the curve, found by counting squares or integration (beyond Class 9, so questions stay with straight lines).

Equations of Motion and Uniform Circular Motion

The Three Equations of Motion (uniform acceleration)

Valid only when acceleration a is constant. Symbols: u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

1. First equation (velocity–time): v = u + at
2. Second equation (position–time): s = ut + ½at²
3. Third equation (position–velocity): v² = u² + 2as

Derivations (graphical, in words)

• From a v–t straight-line graph, acceleration = slope = (v − u)/t. Rearranging gives v = u + at.
• Distance = area under the v–t graph = area of rectangle (ut) + area of triangle (½ × t × (v − u)). Substituting (v − u) = at gives s = ut + ½at².
• Distance = area of trapezium = ½(u + v)t. Substituting t = (v − u)/a and simplifying gives v² = u² + 2as.

Tip: For a body that starts from rest, u = 0, so the equations simplify to v = at, s = ½at², v² = 2as. For free fall, replace a with g = 9.8 m/s² and s with height h.

Uniform Circular Motion

When an object moves along a circular path with constant speed, its motion is called uniform circular motion.

• Even though the speed is constant, the velocity changes continuously because the direction of motion changes at every point (velocity is a vector).
• Since velocity changes, uniform circular motion is an accelerated motion.
• The direction of velocity at any point is along the tangent to the circle.
• Speed in a circle: v = (2πr) / t, where r = radius and t = time for one revolution (circumference 2πr).

Examples: the Moon revolving around the Earth, tip of a clock's second hand, an athlete on a circular track, a stone whirled in a horizontal circle by a string.

Trap: A common error is saying uniform circular motion has zero acceleration because speed is constant. It is accelerated because the direction (hence velocity) keeps changing.

Frequently asked questions

Are these Motion notes free?

Yes — the Motion notes for Science (Class 9) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Motion notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Motion chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Motion.