Motion

An object is said to be in motion if its position changes with time with respect to a reference point (also called an origin). If its position does not change, it is at rest. Rest and motion are relative — a passenger is at rest with respect to a fellow passenger but in motion with respect to a tree on the road, so you must always name the reference point when you describe motion.

To describe motion along a straight line we use two related but different quantities. Distance is the actual length of the path covered by the object, whatever its shape — it is a scalar quantity, so it needs only a magnitude, no direction, and it can never decrease with time. Displacement is the shortest distance between the initial and the final position, measured in a straight line, together with its direction — it is a vector quantity.

Two facts the examiner checks often: (i) the magnitude of displacement is always less than or equal to the distance travelled — the two are equal only when the object moves along a straight line without changing direction; (ii) displacement can be zero even when the distance covered is not — an athlete who runs one full round of a circular track returns to the starting point, so distance = circumference of the track but displacement = 0.

Different objects can take different times to cover the same distance — some move faster, some slower. Speed tells us how fast an object is moving; it is defined as the distance covered by the object in unit time. Speed is a scalar quantity with SI unit metre per second (m/s or m s⁻¹); kilometre per hour (km/h) is also common.

An object moving equal distances in equal intervals of time is said to move with uniform speed; if it covers unequal distances in equal intervals of time, its speed is non-uniform (variable). Most everyday motion — a bus in traffic, a falling leaf — is non-uniform, so we describe it using average speed: the total distance travelled divided by the total time taken. Average speed gives one overall number even though the speed may have changed moment to moment during the journey; for truly uniform motion, average speed equals the actual speed at every instant.

The odometer of a vehicle measures the distance moved, and the speedometer shows the speed at that instant. Distinguishing 'speed at an instant' from 'average speed over the whole trip' is a common exam trap — read the question carefully to see which one is being asked.

Speed alone does not tell us the direction an object is moving in, so it does not fully describe motion. Velocity is the displacement of an object in unit time, i.e. speed in a given direction — it is a vector quantity, with the same SI unit as speed, m/s.

An object has uniform velocity if it moves in a straight line and its displacement is the same in equal intervals of time, in the same direction throughout — both magnitude and direction stay constant. Its velocity is variable (non-uniform) if either its speed changes, or its direction changes, or both change — even if the object moves at constant speed along a curved path, as in uniform circular motion.

Just as we average speed, we use average velocity for motion with changing velocity: total displacement divided by total time taken. A very useful special-case formula for numericals: when an object moves along a straight line with uniform acceleration, its average velocity equals the arithmetic mean of the initial velocity u and the final velocity v, i.e. average velocity = (u + v)/2. This formula applies only for uniform acceleration; for non-uniform acceleration you must use total displacement/total time.

The velocity of an object need not stay the same during motion — it may increase, decrease or change direction. Acceleration is the rate of change of velocity with time: a = (v − u)/t, where u is the velocity at the start of the interval, v is the velocity at the end, and t is the time taken. Acceleration is a vector; its SI unit is metre per second squared (m/s² or m s⁻²).

If the velocity changes at a uniform rate, the acceleration is uniform — equal changes in velocity occur in equal intervals of time, as in the motion of a freely falling body. If the rate of change of velocity is not constant, the acceleration is non-uniform, as in a car being driven through city traffic.

Acceleration is positive when velocity increases in the direction of motion, and negative (called retardation or deceleration) when velocity decreases in the direction of motion. A common trap: acceleration is not zero just because speed is constant — in uniform circular motion the speed stays constant but the direction keeps changing, so the velocity is changing and the motion is accelerated.

A distance-time graph is one of the simplest ways to describe motion. Time is plotted on the x-axis and distance on the y-axis, always on a convenient, uniform scale.

When an object moves at uniform speed, it covers equal distances in equal time intervals, so the distance-time graph is a straight line — the steeper the line, the greater the speed. Speed can be read directly from the graph as its slope: for any two points on the line, speed = (s₂ − s₁)/(t₂ − t₁).

When speed is non-uniform, the object covers unequal distances in equal time intervals and the graph is a curved line; different portions of the curve have different slopes, giving the speed at different times.

A distance-time graph that is a horizontal line parallel to the time axis means the distance is not changing while time passes — the object is at rest. Students often confuse this with a line through the origin; remember rest means zero slope, not zero distance. Reading distance-time graphs correctly — slope gives speed, and shape identifies rest, uniform or non-uniform motion — is asked almost every year in some form.

A velocity-time graph plots velocity on the y-axis against time on the x-axis, and carries more information than a distance-time graph because both acceleration and distance travelled can be read from it.

If velocity stays constant, the graph is a straight line parallel to the time axis — the area between this line and the time axis, over an interval, gives the distance covered in that interval.

If the object moves with uniform acceleration, the velocity-time graph is a straight line inclined to the time axis. Its slope gives the acceleration: a = (v − u)/t. A rising line means increasing velocity; a uniformly decelerating object gives a line falling towards the time axis.

For non-uniform acceleration, the velocity-time graph is a curved line.

The single most useful fact for numericals: the area enclosed between the velocity-time graph and the time axis, over a chosen interval, equals the distance (or displacement) covered in that interval — a rectangle for uniform velocity, a triangle or trapezium for uniformly accelerated motion. This area method is exactly how the three equations of motion are derived, so understanding it earns marks in the derivation questions too.

For an object moving with uniform acceleration, three equations connect initial velocity u, final velocity v, acceleration a, time t and distance s. All three can be derived from a velocity-time graph, and CBSE frequently asks for this graphical derivation, so know every step.

First equation, v = u + at: Consider an object with initial velocity u at time t = 0, moving with uniform acceleration a. Its velocity-time graph is the straight line AB, where A is on the velocity axis at height u and B is at time t with velocity v. Draw BC perpendicular to the time axis and AD parallel to the time axis meeting BC at D, so OC = t, OA = BD = u and BC = v.

From the graph, BC = BD + DC, i.e. v = u + DC ... (i)

The slope of line AB gives the acceleration: a = BD/AD = DC/OC (since AD = OC = t), so DC = at.

Substituting in (i): v = u + at, the first equation of motion. It connects velocity to time directly and is the equation to use whenever a question does not mention distance.

Second equation, s = ut + ½at²: Using the same velocity-time graph, the distance travelled s equals the area under line AB between t = 0 and t = t, which is the area of rectangle OADC plus the area of triangle ABD.

Area of rectangle OADC = OA × OC = u × t. Area of triangle ABD = ½ × AD × BD = ½ × t × at (since BD = at from the first equation). So s = ut + ½at², the second equation of motion, used whenever time is known but final velocity is not asked directly.

Third equation, v² = u² + 2as: Again from the same graph, s = area of trapezium OABC = ½ × (sum of parallel sides) × height = ½ × (OA + CB) × OC = ½(u + v)t.

From the first equation, t = (v − u)/a. Substituting: s = ½(u + v) × (v − u)/a = (v² − u²)/2a, which rearranges to v² = u² + 2as. Reach for this equation when time is not given at all. Learn all three derivations together — a five-mark question often asks for one specific equation, and marks are lost for skipping the labelled-graph step, so always draw and label the graph first.

When an object moves along a circular path at constant speed, its motion is called uniform circular motion. This sounds like it should not be 'accelerated' motion because the speed never changes, but it is — because velocity is a vector, and the direction of motion keeps changing continuously as the object goes around the circle, even though its magnitude (speed) stays fixed. A changing velocity means there is acceleration, directed towards the centre of the circle.

Common everyday examples asked in exams: an athlete running on a circular track, the tip of the second, minute or hour hand of a clock, the moon revolving around the Earth, the blades of an electric fan or a mixer grinder, and a cyclist going around a circular park at constant speed.

If an object moves along a circle of radius r and completes one round in time t (the time period), the distance covered in one round is the circumference of the circle, 2πr, so the speed is: speed = 2πr/t. This is the standard formula for uniform circular motion numericals — use the correct value of π (22/7 or 3.14) and keep radius and circumference in the same unit as the answer required.