Hello, and welcome to your physics lesson for today. We are diving into the fascinating world of motion — specifically, motion in one dimension. By the end of this lesson, you will understand the difference between scalars and vectors, how to describe motion using distance, displacement, speed, and velocity, and how to analyse accelerated motion through graphs and equations. Let us begin.
First, we need to understand how we describe physical quantities. Some quantities need only a number and a unit to be fully described. These are called scalar quantities, or simply scalars. Mass, length, time, distance, speed, temperature, and energy are all scalars. If I tell you a box has a mass of five kilograms, you know everything you need to know.
But other quantities require something more — they need direction. These are called vector quantities, or vectors. Displacement, velocity, acceleration, force, and momentum are all vectors. If I say a car moved fifty metres, you would immediately ask, "In which direction?" That is why displacement, which is the shortest distance from initial to final position in a specific direction, is a vector. Vectors are written with an arrow above the symbol, like v⃗, or in bold. A negative sign simply means the opposite direction.
Now, let us talk about motion itself. Nothing in the universe is truly at rest in an absolute sense. The Earth spins, it orbits the Sun, and our entire galaxy moves through space. However, in everyday terms, we say a body is at rest when it does not change its position with respect to its immediate surroundings. A book on your desk is at rest relative to the desk. A body is in motion when its position changes with respect to those surroundings.
When a body moves along a straight line, we call this one-dimensional motion, or rectilinear motion. Think of a train on a straight track, or a stone falling vertically downward. We can represent this motion by plotting position along an x-axis that follows the path of motion.
Here is where distance and displacement become crucially different. Distance is the total length of the path travelled — it is a scalar, always positive, and depends on the route taken. Displacement is the straight-line change in position from start to finish, complete with direction — it is a vector.
Imagine walking four metres east, then three metres north. Your distance travelled is seven metres. But your displacement is five metres, directed at an angle of about thirty-seven degrees north of east. The magnitude of displacement is never greater than the distance, though they can be equal if the path is perfectly straight. Most importantly, displacement can be zero even when distance is not — walk in a complete circle back to your starting point, and your displacement vanishes while your distance equals the full circumference.
Speed and velocity follow the same pattern. Speed is the rate of change of distance with time — a scalar. Velocity is the rate of change of displacement with time — a vector.
We write the relation as: speed equals distance divided by time.
Or, v = s/t, where s represents distance in metres and t represents time in seconds.
The SI unit is m s⁻¹.
Uniform speed means equal distances in equal time intervals. Non-uniform speed means the distance covered varies from interval to interval. For non-uniform motion, we speak of instantaneous speed — the speed at a particular moment — and average speed, which is total distance divided by total time.
Velocity adds direction to this picture. Uniform velocity requires equal displacements in equal times, with no change in direction. Even if speed stays constant, velocity changes if direction changes — as in circular motion, where the object continuously turns even at steady speed.
Now we reach one of the most important concepts: acceleration. Acceleration is the rate of change of velocity with time.
Acceleration equals change in velocity divided by time taken.
Or, a = (v − u)/t, where u is initial velocity in m s⁻¹, v is final velocity in m s⁻¹, a is acceleration in m s⁻², and t is time in seconds.
Rearranging, we get the first equation of motion: v = u + at.
When velocity increases, acceleration is positive. When velocity decreases, we call this retardation or deceleration — it is negative acceleration.
Uniform acceleration means equal changes in velocity in equal time intervals. A freely falling body under gravity experiences uniform acceleration of approximately 9.8 m s⁻², often rounded to 10 m s⁻² for simplicity. This acceleration due to gravity, denoted g, is independent of the mass of the falling object.
Graphs bring these concepts to life visually. In a displacement-time graph, time runs along the horizontal axis and displacement along the vertical. A horizontal line means the body is stationary. A straight inclined line means uniform velocity — and the slope of that line gives the velocity directly. A curved line indicates variable velocity, and the slope of the tangent at any point gives the instantaneous velocity.
Positive slope means motion away from the origin; negative slope means motion back toward it.
The velocity-time graph is equally powerful. Here, the slope gives acceleration. A horizontal line means constant velocity and zero acceleration. A straight inclined line means uniform acceleration. The area under the curve — between the graph and the time axis — gives the displacement. Areas above the time axis count as positive displacement; areas below count as negative.
For example, a vehicle accelerating uniformly from rest to eighty metres per second in eight seconds traces a triangle on the velocity-time graph. The area of that triangle — one-half times base times height — equals three hundred twenty metres, which is the displacement. The slope — eighty metres per second divided by eight seconds — gives ten metres per second squared as the acceleration.
The acceleration-time graph completes the picture. For uniform velocity, this graph is a horizontal line at zero. For uniform acceleration, it is a horizontal line at a constant positive value. For uniform retardation, it is a horizontal line at a constant negative value. The area under this graph gives the change in velocity.
Now we derive the three equations of uniformly accelerated motion. These relationships connect initial velocity, final velocity, acceleration, time, and displacement.
First, from the definition of acceleration: acceleration equals change in velocity divided by time taken.
This gives us: v = u + at.
Equivalently, a = (v − u)/t.
This is our first equation.
Second, the displacement equals average velocity multiplied by time. Average velocity for uniformly accelerated motion is (u + v)/2, the arithmetic mean of initial and final velocities. Substituting our first equation and simplifying: displacement equals average velocity times time, which becomes initial velocity times time plus half acceleration times time squared. This gives us our second equation: s = ut + ½at². Here, s is displacement in metres, u is initial velocity in m s⁻¹, t is time in seconds, and a is acceleration in m s⁻².
Third, eliminating time between the first two equations gives: final velocity squared equals initial velocity squared plus two times acceleration times displacement. This is our third equation: v² = u² + 2as, remarkably useful when time is unknown.
Here, v is final velocity in m s⁻¹, u is initial velocity in m s⁻¹, a is acceleration in m s⁻², and s is displacement in metres.
For a body starting from rest, u = 0, these simplify to: v = at, s = ½at², and v² = 2as.
We can also write the second equation as s = ½(u + v)t, which is particularly useful when acceleration is unknown.
Let us apply these equations. Imagine a car starts from rest and accelerates uniformly at two metres per second squared for ten seconds. Its final velocity is v = u + at = 0 + 2 × 10 = 20 m s⁻¹. The distance covered is s = ut + ½at² = 0 × 10 + ½ × 2 × 10² = 100 metres.
Or consider a ball thrown upward at twenty metres per second. Gravity provides a retardation of 10 m s⁻². At the highest point, velocity becomes zero. Using v = u − gt, where g equals 10 m s⁻², the time to reach the top is t = u/g = 20/10 = 2 seconds. The maximum height, from s = ut − ½gt², is 20 × 2 − ½ × 10 × 2² = 40 − 20 = 20 metres.
Let us recap the essential points.
First, scalars have magnitude only; vectors have magnitude and direction. Distance and speed are scalars; displacement, velocity, and acceleration are vectors.
Second, motion is relative — rest and motion depend on your frame of reference. One-dimensional motion occurs along a straight line.
Third, the slope of a displacement-time graph gives velocity; the slope of a velocity-time graph gives acceleration.
Fourth, the area under a velocity-time graph gives displacement.
Fifth, the equations of motion — v = u + at, s = ut + ½at², v² = u² + 2as, and s = ½(u + v)t — completely describe uniformly accelerated motion.
Sixth, freely falling bodies experience uniform acceleration due to gravity, approximately 9.8 m s⁻² downward.
Physics is not merely about memorising formulas — it is about seeing patterns in how objects move through space and time. The tools you have learned today will serve you throughout your study of mechanics. Keep practising, stay curious, and remember: every complex motion can be broken down into these fundamental principles. Until next time, keep exploring the world of physics.