Rest
An object is said to be at rest when its position does not change with time with respect to a reference point. Rest and motion are relative — an object can be at rest for one observer and in motion for another.
Motion
An object is in motion when its position changes with time in the frame of reference of the observer. Motion can be uniform or non-uniform, along a straight line or curved path.
Distance
The total length of the path covered by an object in a given time interval. It is a scalar quantity — only magnitude, no direction. Distance is always positive or zero, never negative.
Displacement
The shortest distance from the initial to the final position of an object. It is a vector quantity — has both magnitude and direction. Displacement can be positive, negative, or zero.
Speed
The distance covered per unit time. It is a scalar quantity. Average speed = total distance ÷ total time. Speed is always positive. SI unit: m/s.
Velocity
Speed with direction. Displacement covered per unit time. It is a vector quantity. Velocity can be positive, negative, or zero. Average velocity = total displacement ÷ total time.
Acceleration
The rate of change of velocity. a = (v − u) / t. Positive if velocity increases; negative (retardation/deceleration) if velocity decreases. SI unit: m/s². It is a vector quantity.
Uniform Motion
An object covers equal distances in equal time intervals. Velocity is constant. Acceleration is zero. Distance-time graph is a straight line.
Non-Uniform Motion
An object covers unequal distances in equal time intervals. Velocity changes. Acceleration may be constant or variable. A car in city traffic is a typical example.
Distance vs Displacement — Key Comparison
Real-World Connection — Odometer vs GPS
Your car's odometer measures distance — it adds up every metre of road covered, including all turns and detours. A GPS showing straight-line distance from start to end measures displacement. On a winding mountain road, the odometer may show 150 km while GPS displacement is only 80 km — this gap is distance vs displacement in real life!
Did You Know? — Everything is Relative!
There is no such thing as absolute rest or absolute motion. The passengers in a moving train are at rest relative to each other, but in motion relative to someone standing outside. Even the "stationary" ground beneath your feet is moving — Earth rotates at ~1,670 km/h and orbits the Sun at ~107,000 km/h. Rest and motion only make sense when you specify relative to what.
JEE Level — Advanced Concepts
These go beyond Class 9 to include vector treatment, instantaneous quantities, variable acceleration, and advanced problem-solving tools needed for JEE.
Instantaneous Velocity JEE
The velocity at a specific instant: v = dx/dt (derivative of position with respect to time). The slope of the tangent to the position-time graph at any point gives instantaneous velocity. The word "velocity" alone implies instantaneous velocity.
Instantaneous Speed JEE
The magnitude of instantaneous velocity: v = |dx/dt| = ds/dt. The slope of the distance-time (not displacement-time) graph gives instantaneous speed. Instantaneous speed is always ≥ 0.
Instantaneous Acceleration JEE
a = dv/dt = d²x/dt². The slope of the tangent to the v-t graph at any point. Also equals the second derivative of position w.r.t. time. Acceleration can be non-zero even when velocity is zero (ball at peak).
Variable Acceleration JEE
When acceleration depends on time, position, or velocity, standard equations (v=u+at) don't apply. Use integration: v = u + ∫a(t)dt or the chain rule: a = v·dv/dx for position-dependent acceleration.
Average vs Instantaneous JEE
Average speed = total distance/total time. Average velocity = total displacement/total time. These are not equal unless object moves in one direction only. For uniform motion, instantaneous = average velocity = instantaneous speed.
Relative Motion JEE
Velocity of A relative to B: v_AB = v_A − v_B. Key for problems involving two moving objects — trains approaching/separating, swimmer in river, etc. Always define positive direction carefully.
JEE Key Result — Average Speed Special Cases
• If a particle travels equal distances at speeds v₁ and v₂: average speed = 2v₁v₂/(v₁+v₂) (harmonic mean)
• If a particle travels equal time intervals at speeds v₁ and v₂: average speed = (v₁+v₂)/2 (arithmetic mean)
• Average speed can never be zero (unless at rest throughout). Average velocity can be zero even if average speed ≠ 0.
A line parallel to the time-axis. Distance doesn't change → object is not moving. Slope = 0 → speed = 0.
A straight line with positive slope. Equal distances in equal times. Slope = constant speed. Steeper line = faster speed.
A curved (non-linear) line. Slope changes → speed is changing. Steepening curve = acceleration; flattening = deceleration.
Parallel to x-axis. Acceleration = 0. The area under this line = distance = velocity × time (a rectangle).
Straight line with positive slope. Slope = constant acceleration. Area under graph = displacement. Area of triangle + rectangle.
Straight line with negative slope. Velocity decreasing. Object is retarding. If it reaches zero, object has stopped.
→ gives speed/velocity
→ gives acceleration
→ gives displacement
→ impossible (infinite speed)
Did You Know? — Train Timetables and s-t Graphs
The distance-time graph of a train between stations looks like a trapezoid: starting from rest (gradual slope), moving at constant speed (steep straight line), then decelerating to stop (decreasing slope). Railway engineers use these graphs to ensure safe acceleration and braking profiles. A vertical line on the graph would mean teleportation — instantaneous travel — which is physically impossible!
JEE Graph Mastery
Beyond basic reading — JEE tests ability to construct graphs, read slopes of tangents, compute areas of irregular shapes, and interpret non-standard graphs like a–x, v–x, and a–v plots.
Area under the a–t graph = change in velocity (Δv). If a is constant: Δv = a × t. For variable acceleration, integrate: Δv = ∫a dt. This is the fundamental link between a-t and v-t graphs.
Used when acceleration depends on position. Use: a = v·dv/dx. The slope of v–x graph multiplied by v gives acceleration. Area under v–x graph has no standard physical meaning unless specified.
Instantaneous velocity at any point = slope of tangent drawn to the s–t curve at that point. Retardation: curve bends toward time-axis (slope decreasing). Acceleration: curve bends away (slope increasing).
If v = b√x, then a = v·dv/dx = b√x · (b/2√x) = b²/2. This is constant acceleration! v² = b²x is a straight line on the v²–x graph, confirming constant acceleration.
JEE Trap — Displacement vs Distance from v–t Graph
The area under the v–t graph gives displacement (net, with sign). To find distance, you must separate regions where v is positive and negative, take absolute values of each area, and add them. If a v–t graph crosses the time axis, the areas above and below axis correspond to motion in opposite directions — don't add them directly for distance!
Final velocity after uniform acceleration
Displacement during uniform acceleration
Velocity after covering a displacement
Shortcut for displacement
Speed in circular motion
Which Equation to Use? — Quick Decision Guide
Know v, u, t — need a or s? → 1st equation (v = u + at) or 2nd (s = ut + ½at²)
No final velocity given? → Use 2nd equation (s = ut + ½at²)
No time given? → Use 3rd equation (v² = u² + 2as)
Object starts from rest? → Set u = 0 in all equations
JEE Additional Formulas
These extend the basic three equations for more complex situations tested at JEE level.
Distance covered in the n-th second only
When a depends on t, x, or v
For a body thrown upward with velocity u
| Quantity | Symbol | SI Unit | Type | Key Formula / Notes |
|---|---|---|---|---|
| Position / Displacement | s, x | m (metre) | Vector | Displacement = final − initial position. Can be +ve, −ve, or zero. |
| Distance | d | m (metre) | Scalar | Total path length. Always ≥ |displacement|. Always positive. |
| Speed | v | m/s | Scalar | v = d/t. Also expressed as km/h. 1 km/h = 5/18 m/s. 1 m/s = 18/5 km/h. |
| Velocity | v | m/s | Vector | v = displacement/time. Can be +ve, −ve, or zero. Direction matters! |
| Acceleration | a | m/s² | Vector | a = (v−u)/t. Negative a = deceleration. CGS unit: cm/s². Dimension: [M⁰L¹T⁻²] |
| Time | t | s (second) | Scalar | Always positive in kinematic problems. Dimension: [T]. |
| Gravitational acceleration (g) | g | m/s² | Vector | g = 9.8 m/s² (precise) or 10 m/s² (problems). Directed downward always. |
| Angular Velocity | ω | rad/s | Vector | For circular motion: v = rω. ω = 2π/T = 2πf. |
Did You Know? — The 5/18 Trick
Converting km/h to m/s: multiply by 5/18. Converting m/s to km/h: multiply by 18/5 = 3.6. Why? Because 1 km = 1000 m and 1 hour = 3600 s, so 1 km/h = 1000/3600 = 5/18 m/s. This conversion is needed in almost every motion problem — memorise it as a reflex! Quick check: 72 km/h = 72 × (5/18) = 20 m/s.
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Train Passengers — Relative Motion
Passengers in a moving train are at rest relative to each other (same frame) but moving relative to someone on the platform. This perfectly illustrates that rest and motion are always relative to a reference frame — there is no absolute rest. -
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Usha Swimming — Speed vs Velocity
Usha covers 180 m in a 90 m pool (goes and returns) in 1 minute. Average speed = 3 m/s but average velocity = 0 m/s because displacement = 0 (she ends where she started). This is the key difference: speed depends on path, velocity depends on displacement. -
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Falling Apple — Free Fall and g
Every object dropped near Earth's surface accelerates at g = 9.8 m/s² regardless of mass. A 1 kg apple and a 10 kg watermelon dropped from the same height hit the ground simultaneously (if air resistance is ignored). This was proved by Galileo and forms the basis of free fall equations. -
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Car Braking — Stopping Distance and v²
Using v² = u² + 2as with v = 0: stopping distance s = u²/2a. Double the speed → 4× the stopping distance! This is why speed limits are so critical — going from 40 km/h to 80 km/h makes you need 4× the distance to stop. Traffic safety laws are literally built on this equation. -
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Moon's Orbit — Uniform Circular Motion
The Moon moves around Earth at nearly constant speed (~1022 m/s) but in a circular path. Its speed is constant but its velocity changes every instant (direction changes). It is therefore accelerating — centripetally, toward Earth. Gravity provides this centripetal force. This is uniform circular motion in the grandest scale! -
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Lightning and Thunder — Calculating Distance
You see lightning instantly (light travels at 3×10⁸ m/s) but hear thunder seconds later (sound travels at ~346 m/s). The time gap × 346 gives the distance to the lightning. Count 3 seconds after the flash? The lightning was about 1 km away. This is real-world application of s = vt. -
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Hammer Throw — Tangential Release
An athlete whirling a hammer in a circle and releasing it experiences uniform circular motion followed by projectile motion. At the moment of release, the hammer flies off tangentially (in a straight line) — exactly the direction it was moving at that instant. This is why Newton's First Law predicts the release direction perfectly. -
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Reaction Time — Safety Distance While Driving
Before a driver hits the brakes, their reaction time (typically 0.2 s) means the car travels "free" distance = speed × 0.2. At 72 km/h (20 m/s), this is 4 metres before braking even starts! Total stopping distance = reaction distance + braking distance. This is why drunk/sleepy driving is deadly — reaction time increases 3–5×.
Did You Know? — Galileo's Ramp Experiment
Galileo couldn't measure free fall directly (too fast for 16th century timing). So he slowed it down by rolling balls down inclined planes and proved that distance is proportional to t² — the same relationship as free fall, just scaled. He then extrapolated to 90° (vertical) to derive g. His experiment with the Leaning Tower of Pisa (dropping two cannonballs) showed mass doesn't affect acceleration — overturning 2000 years of Aristotelian physics in one afternoon.
✅ Air Resistance Ignored
Unless stated, all free fall problems assume no air resistance. In reality, a feather falls slower than a stone — but in a vacuum they fall at the same rate. This is a standard simplifying assumption.
✅ Equations Valid Only for Constant a
The three equations of motion (v=u+at, s=ut+½at², v²=u²+2as) are only valid when acceleration is constant. For variable acceleration, they cannot be used directly.
✅ g is Approximately Constant
Near Earth's surface, g ≈ 9.8 m/s² and is treated as constant. For very large heights, g decreases with altitude. School problems use 10 m/s² for simplicity.
⚠️ Distance ≠ Displacement
The most common mistake: using distance in velocity formula (should be displacement) or vice versa. Average speed uses total distance; average velocity uses net displacement.
⚠️ Speed ≠ Velocity
Speed is scalar (only magnitude), velocity is vector (magnitude + direction). A car going around a circular track at constant speed still has changing velocity because direction changes.
⚠️ v=0 Does Not Mean a=0
A ball thrown upward has v=0 at the highest point but a = −g = −9.8 m/s² (or −10 m/s²) at that point. The acceleration due to gravity never becomes zero just because velocity does.
📌 Sign Convention is Key
Always define a positive direction first! Upward = +ve usually. Then g = −10 m/s² (acts downward). Consistency is everything — a wrong sign flips your entire answer.
📌 Uniform Circular Motion
Speed is constant, but velocity is not — direction keeps changing. Therefore the object IS accelerating (centripetally, toward the centre), even though its speed is constant.
JEE Advanced Notes
Additional precision and traps specific to JEE-level problems.
JEE Trap — Sₙ Formula JEE
Sₙ = u + a/2(2n−1) gives displacement in the nth second. It's NOT Sₙ = u·n + ½a·n² (which gives total displacement in n seconds). Very common mistake in MCQs.
Avg Speed ≥ |Avg Velocity| JEE
Always true: average speed ≥ |average velocity|. They are equal only when object moves in one direction without reversing. If direction reverses: distance > |displacement| → speed > |velocity|.
Instantaneous Speed = |Instantaneous Velocity| JEE
At any instant, instantaneous speed = magnitude of instantaneous velocity. This is NOT true for averages. Keep this clearly separate in your mind.
Galileo's Odd Number Ratios JEE
Body from rest with constant a: distances in 1st, 2nd, 3rd... seconds are in ratio 1:3:5:7... (odd numbers). Total distances in 1s, 2s, 3s... are 1:4:9:16... (squares). Frequently tested in multiple-correct JEE questions.
Free Fall: Ascent = Descent Time JEE
Time of ascent = time of descent = u/g (if no air resistance). Speed at projection = speed on return. Total time T = 2u/g. The body is symmetric in time — a key property used in many problems.
Retardation Sign Trap JEE
Retardation is deceleration, meaning acceleration is opposite to velocity. If object moves in +ve direction and decelerates, a is negative. Don't confuse "retardation of 5 m/s²" with "a = +5" — it means a = −5 m/s² in that direction.
🏃 1st Equation: v = u + at
📏 2nd Equation: s = ut + ½at²
⚡ 3rd Equation: v² = u² + 2as
🌍 Free Fall / Projectile Up
🔄 Uniform Circular Motion
Try These Classic Problems!
P1 (Braking car): Car at 72 km/h (= 20 m/s) brakes with a = −5 m/s². Stopping distance? → Use 3rd Eq: s = (0² − 20²) / (2×−5) = 40 m
P2 (Free fall): Ball dropped from 20 m. Time to hit ground? → s=½gt²: 20=½×10×t² → t = 2 s, v = gt = 20 m/s
P3 (Train): Starts from rest, reaches 20 m/s in 5 min (300 s). Acceleration? → a = (20−0)/300 = 1/15 m/s²