Kinematics - Motion in One and Two Dimensions

Why Kinematics Is a Sure-Shot Scorer

Open any past NDA General Ability paper and you will spot two or three questions on motion — a falling stone, a car braking, a ball thrown up off a cliff. These are formula-driven rather than memory-based, so once you know the handful of rules you can solve them in under a minute and bank the marks.

Kinematics also sits quietly under everything that follows in mechanics. The laws of motion, work-energy theorems and gravitation all assume you already understand velocity and acceleration. If those basics are shaky, every later chapter feels confusing. Master kinematics early and the rest of physics becomes far easier and far less scary.

The good news for a Class 11-12 student is that there is very little to memorise — just three equations, a couple of graph rules and the projectile formulas. The skill lies in reading the question carefully, listing what is given, and choosing the right tool. We will practise exactly that throughout this page.

Scalars and Vectors: The First Sorting

Every quantity in motion is either a scalar or a vector. A scalar has only size (magnitude). A vector has size and direction.

• Scalars: distance, speed, time, mass, temperature.
• Vectors: displacement, velocity, acceleration, force.

This single distinction decides whether a sign (+ or −) matters in your working. For a ball thrown straight up, taking up as positive means gravity acts as a negative acceleration. Forget the sign and your whole answer collapses, even if every number is correct.

Vectors can also be added in a special way. Two velocities pointing in the same direction simply add up, but velocities at an angle must be combined using the parallelogram or triangle rule. This is exactly why a swimmer crossing a flowing river ends up downstream of the point they aimed for — their own velocity and the river's velocity combine into a single resultant vector.

Distance vs Displacement

Distance is the total path length actually travelled. It is a scalar and can never decrease — it only piles up.

Displacement is the straight-line gap from the start point to the end point, with direction. It is a vector and can be zero even after a long journey.

Example: walk 4 m east, then 3 m north. Distance = 4 + 3 = 7 m. Displacement = √(4² + 3²) = 5 m, north-east. Run one full lap of a 400 m track and your distance is 400 m but your displacement is zero, because you finish exactly where you started.

This idea catches students out again and again. A question may describe a long, winding journey just to test whether you notice that the start and end points are close together. Always pause and ask: are they asking for the actual path covered, or the net shift in position? The word “total path” signals distance, while “how far from the start” signals displacement.

Speed, Velocity and Acceleration

Speed = distance ÷ time (scalar). Velocity = displacement ÷ time (vector). A car can move at a steady speed of 40 km/h yet have changing velocity if it keeps turning, because direction is changing.

Acceleration is the rate of change of velocity. If velocity rises, a is positive; if velocity falls (braking), a is negative and is called retardation or deceleration. Uniform motion means constant velocity, so a = 0.

It is worth pausing on one subtle point that examiners love. Because velocity is a vector, an object can be accelerating even at constant speed. A car going around a roundabout at a steady 40 km/h is constantly accelerating, because its direction — and therefore its velocity — keeps changing. Acceleration responds to any change in velocity, whether that change is in size, in direction, or in both.

Also keep your units tidy. Speed and velocity are measured in metres per second (m/s), while acceleration is metres per second every second, written m/s². If an answer comes out in the wrong units, you have almost certainly slipped somewhere in the algebra.

The Three Equations of Motion

When acceleration is uniform (constant), three equations connect initial velocity u, final velocity v, acceleration a, time t and displacement s.

How to choose? Look at what is missing in the question:

• No displacement given → use v = u + at.
• No final velocity given → use s = ut + ½at².
• No time given → use v² = u² + 2as.

Free Fall Under Gravity

Free fall is just uniformly accelerated motion with a = g, the acceleration due to gravity, about 9.8 m/s² (often rounded to 10 m/s² in NDA problems).

Replace a with g in the three equations. Take downward as positive when something is dropped; take upward as positive when something is thrown up (then g becomes −g).

• Dropped object: u = 0, so v = gt and s = ½gt².
• Thrown up with speed u: it stops momentarily at the top (v = 0), then falls back.

A lovely symmetry hides inside vertical throws. The time taken to rise equals the time taken to come back down, and the speed with which the object returns to the launch height equals the speed it was thrown with. So a ball thrown up at 20 m/s returns to the same height moving at 20 m/s, just in the opposite direction. NDA questions often exploit this symmetry to ask for total time of flight or the speed on return.

Reading Motion Graphs

NDA loves graph-based questions because they test understanding, not just plugging numbers.

Displacement–time graph

• The slope gives velocity.
• A straight slanted line = uniform velocity.
• A horizontal line = object at rest.
• A curve = changing velocity (acceleration).

Velocity–time graph

• The slope gives acceleration.
• The area under the line gives displacement.
• A horizontal line = uniform velocity (zero acceleration).

Projectile Motion in Two Dimensions

A projectile is any body thrown into the air that then moves only under gravity — a cricket ball, a shell from a gun, a long jumper. Its path is a parabola.

The trick is to split the motion into two independent parts:

• Horizontal: constant velocity (no acceleration, ignoring air drag).
• Vertical: uniformly accelerated by g, exactly like free fall.

Range is maximum when θ = 45°, because sin 2θ reaches its peak of 1 there. Two angles that add up to 90° (say 30° and 60°) give the same range, though the steeper angle climbs higher and stays in the air longer.

One more fact worth carrying into the exam hall: at the highest point of its flight, a projectile still has its full horizontal velocity (u cosθ), even though its vertical velocity has dropped to zero. The body is not at rest at the top — it is merely at the top of its vertical climb while continuing to drift sideways. Students who assume the projectile “stops” at the peak get caught out every time.

Uniform and Non-Uniform Motion

Motion comes in two broad flavours, and telling them apart decides which formulas you may use.

Uniform motion

The body covers equal distances in equal intervals of time. Its velocity is constant and its acceleration is zero. On a displacement–time graph this shows up as a straight slanted line. A train cruising at a fixed 60 km/h on a straight track is in uniform motion.

Non-uniform motion

The body covers unequal distances in equal intervals of time, so its velocity keeps changing and acceleration is not zero. A bus pulling out of a stop, a stone falling, a sprinter starting a race — all are non-uniform.

Closely linked is the idea of relative velocity. The velocity of one body as seen from another is found by subtracting their velocity vectors. Two trains moving in the same direction at 50 and 30 m/s have a relative speed of 20 m/s, but if they move towards each other the relative speed becomes 80 m/s. This is why an oncoming train flashes past so quickly.

Worked Example: A Braking Car

The negative sign on a simply confirms the car is slowing down. The stopping distance is 50 m.

Common Mistakes to Avoid

• Mixing up distance and displacement when a body returns to its start.
• Forgetting the sign of acceleration in upward throws and braking.
• Using the constant-acceleration equations when acceleration is not constant.
• Mixing units — convert km/h to m/s by multiplying by 5/18 before substituting.

Previous-Year Style Question

Notice the pattern: “dropped” tells you u = 0, “height” tells you to find s, and time is given — so the second equation of motion is the natural pick.

Quick Recap and Revision

Drill five or six numerical problems daily, always writing down the “given” list first. That habit alone will turn kinematics into easy, guaranteed marks in your NDA paper.