Everything around you is moving — even when it looks still! Motion is simply the change in position of an object with time. This chapter teaches you to describe, measure and predict that motion using clean equations.
Rest & Motion
An object is in motion if its position changes with time relative to a reference point.
Speed vs Velocity
Speed is how fast (scalar); velocity is how fast and in which direction (vector).
Acceleration
The rate at which velocity changes with time — speeding up, slowing down or turning.
Equations of motion
Three neat formulae link u, v, a, t and s for uniform acceleration.
1. Motion is relative
An object is said to be in motion if it changes its position with respect to a fixed point called the reference point or origin. A passenger sitting in a moving train is at rest with respect to a fellow passenger but in motion with respect to a person standing on the platform. So whether an object is at rest or in motion depends on the observer. This is why we always describe motion with respect to a chosen reference point. The simplest motion to study is motion along a straight line, called rectilinear motion.
2. Distance and displacement
Distance is the total length of the path travelled by an object, regardless of direction. It is a scalar quantity — it has only magnitude. Displacement is the shortest straight-line distance from the initial to the final position, measured in a particular direction. It is a vector quantity — it has both magnitude and direction. If an athlete runs once around a circular track of 200 m and comes back to the start, the distance covered is 200 m but the displacement is zero, because the starting and finishing points are the same. Displacement can be zero, positive or negative, but distance is always positive (or zero) and can never be less than the magnitude of displacement.
3. Uniform and non-uniform motion
An object is in uniform motion when it covers equal distances in equal intervals of time, however small the time intervals may be. For example, a car moving steadily at 50 km/h covers the same distance each second. In non-uniform motion, the object covers unequal distances in equal intervals of time — like a car in city traffic that speeds up and slows down. Most real motion is non-uniform.
4. Speed
Speed is the distance travelled by an object per unit time. Speed = distance ÷ time. Its SI unit is metres per second (m/s or m s-1). Speed is a scalar. When an object covers different distances in different times, we use average speed = total distance ÷ total time. Average speed tells us the overall rate of motion but hides the details of speeding up and slowing down.
5. Velocity
Velocity is the speed of an object in a given direction, that is, the displacement per unit time. Velocity = displacement ÷ time. Its SI unit is also m/s, but it is a vector. The velocity of an object can change by changing its speed, its direction of motion, or both. Average velocity for uniformly changing velocity is the average of the initial velocity (u) and the final velocity (v): average velocity = (u + v) ÷ 2.
6. Acceleration
Acceleration measures how quickly the velocity of an object changes. Acceleration = change in velocity ÷ time taken = (v − u) ÷ t. Its SI unit is metres per second squared (m/s2 or m s-2) and it is a vector. If velocity increases, acceleration is positive; if velocity decreases, acceleration is negative and is often called retardation or deceleration. When the change in velocity is equal in equal intervals of time, the motion is said to be uniformly accelerated. The free fall of a body near the Earth is an example of uniformly accelerated motion (a = g = 9.8 m/s2).
7. Graphical representation of motion
Graphs let us picture motion at a glance. In a distance-time graph, the slope (steepness) of the line gives the speed. A straight, slanting line means uniform speed; a horizontal line means the object is at rest; a curved line means non-uniform speed. In a velocity-time graph, the slope gives the acceleration, and the area enclosed under the line (between the line and the time axis) gives the displacement (distance travelled). A horizontal line in a velocity-time graph means constant velocity (zero acceleration), while a slanting straight line means uniform acceleration.
8. Equations of motion by graphical method
For an object moving with uniform acceleration, three equations connect the initial velocity (u), final velocity (v), acceleration (a), time (t) and displacement (s). They are derived from the velocity-time graph: the first from the definition of acceleration, the second from the area under the graph, and the third by combining the two. These equations work only when acceleration is constant. They are the most important tools of this chapter and are used to solve almost every numerical.
9. Uniform circular motion
When an object moves along a circular path at a constant speed, its motion is called uniform circular motion. Even though the speed is constant, the direction of motion changes continuously at every point of the circle. Since velocity depends on direction, the velocity keeps changing — therefore uniform circular motion is an accelerated motion. The speed of an object moving in a circle of radius r in time T (one full round) is v = 2πr ÷ T. Examples include a stone whirled on a string, the tip of a clock’s second hand, and the Moon revolving around the Earth.
- Speed = distance ÷ time; SI unit m/s
- Average speed = total distance ÷ total time
- Velocity = displacement ÷ time (vector)
- Acceleration a = (v − u) ÷ t; SI unit m/s2
- First equation: v = u + at
- Second equation: s = ut + ½at2
- Third equation: v2 = u2 + 2as
- Speed in circular motion: v = 2πr ÷ T
- Distance-time graph slope = speed; velocity-time graph slope = acceleration, area = displacement
A car starts from rest and attains a velocity of 20 m/s in 5 s. Find (a) its acceleration and (b) the distance it travels in this time.
- List the data: u = 0 (starts from rest), v = 20 m/s, t = 5 s.
- Use the first equation v = u + at to find a: 20 = 0 + a × 5.
- So a = 20 ÷ 5 = 4 m/s2.
- Now use the second equation s = ut + ½at2 = 0 × 5 + ½ × 4 × 52.
- s = ½ × 4 × 25 = 50 m.
A bus moving at 36 km/h is brought to rest in 10 s by applying brakes. Find the retardation and the distance covered before stopping.
- Convert speed to SI units: 36 km/h = 36 × (1000 ÷ 3600) = 10 m/s. So u = 10 m/s.
- The bus stops, so final velocity v = 0; time t = 10 s.
- Use v = u + at: 0 = 10 + a × 10, giving a = −1 m/s2 (negative means retardation of 1 m/s2).
- Find distance using v2 = u2 + 2as: 0 = 102 + 2 × (−1) × s.
- 0 = 100 − 2s, so 2s = 100 and s = 50 m.
Remember the three equations as “Vat, Suttat, Vu-2as”: v = u + at (no s), s = ut + ½at2 (no v), v2 = u2 + 2as (no t). Each equation is missing exactly one of the five quantities — pick the equation that does NOT contain the quantity you are not given!
The most common mistake is forgetting to convert km/h to m/s before using the equations. Always multiply km/h by 5/18 (i.e. × 1000 ÷ 3600) to get m/s. The second biggest mistake is dropping the negative sign on acceleration during retardation — if the object slows down, a must be negative in the equations.
Q1. Distinguish between distance and displacement with one example.
Answer: Distance is the total length of the actual path travelled and is a scalar (only magnitude); displacement is the shortest straight-line distance from the start to the end point in a definite direction and is a vector (magnitude and direction). Example: if a person walks 4 m east and then 3 m west, the distance covered is 4 + 3 = 7 m, but the displacement is only 1 m towards the east. Displacement can be zero even when distance is not.
Q2. Why is uniform circular motion called an accelerated motion?
Answer: In uniform circular motion the speed of the object stays constant, but its direction of motion changes continuously at every point along the circle. Velocity is a vector that depends on both speed and direction, so a change in direction means the velocity is changing. Since acceleration is the rate of change of velocity, a continuously changing velocity means the object is accelerating. Hence uniform circular motion is an accelerated motion.
Q3. A train starting from rest attains a velocity of 72 km/h in 5 minutes. Assuming uniform acceleration, find the acceleration and the distance travelled.
Answer: u = 0; v = 72 km/h = 72 × 5/18 = 20 m/s; t = 5 min = 300 s. Using v = u + at: 20 = 0 + a × 300, so a = 20 ÷ 300 = 0.067 m/s2 (1/15 m/s2). Distance s = ut + ½at2 = 0 + ½ × (1/15) × 3002 = ½ × (1/15) × 90000 = 3000 m = 3 km.
Q4. What does the slope of a distance-time graph and the area under a velocity-time graph represent?
Answer: The slope (steepness) of a distance-time graph represents the speed of the object; a steeper line means a higher speed, a horizontal line means the object is at rest. In a velocity-time graph, the slope of the line gives the acceleration, while the area enclosed under the line (between the graph line and the time axis) gives the displacement (distance travelled) by the object in that time interval.
- ✅ Motion is the change of position with time, always measured relative to a reference point.
- ✅ Distance is scalar; displacement is vector and is the shortest path with direction.
- ✅ Speed = distance/time; velocity = displacement/time; acceleration = change in velocity/time.
- ✅ Three equations of motion: v = u + at, s = ut + ½at2, v2 = u2 + 2as (uniform acceleration only).
- ✅ Distance-time slope = speed; velocity-time slope = acceleration, area = displacement.
- ✅ Uniform circular motion has constant speed but changing direction, so it is accelerated.
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