Kinematics

Kinematics describes how objects move without asking why. The first step is to fix a frame of reference — an origin and axes — because position, and therefore motion, is always measured relative to something. A passenger is at rest with respect to the train but moving with respect to the platform; both descriptions are correct.

Two pairs of terms must never be confused in NEET. Distance is the total path length actually covered; it is a scalar and can only increase. Displacement is the straight-line vector from initial to final position; it has direction and can be zero even after a long journey — a runner who completes one full lap has covered a distance equal to the track length but a displacement of zero. The magnitude of displacement is always less than or equal to the distance.

From these come the rate terms. Speed is distance per unit time (a scalar); velocity is displacement per unit time (a vector). Average velocity over an interval is total displacement divided by total time, while instantaneous velocity is the limit of this ratio as the interval shrinks to zero — geometrically, the slope of the tangent to the position-time graph. Because displacement can be small while distance is large, average speed is generally greater than the magnitude of average velocity, and they are equal only for motion in a single straight line without reversal.

Graphs are a NEET favourite. On a position-time graph the slope gives velocity: a straight slanting line means uniform velocity, a curve means changing velocity. On a velocity-time graph the slope gives acceleration and the area under the curve gives displacement. Reading these slopes and areas correctly answers a large share of kinematics questions without any algebra.

Acceleration is the rate of change of velocity, $a = \dfrac{\Delta v}{\Delta t}$, a vector with SI unit $\text{m/s}^{2}$. Because velocity is a vector, acceleration appears whenever speed changes, direction changes, or both — a car rounding a bend at constant speed is still accelerating. When acceleration acts opposite to velocity the body slows down (retardation); a negative sign for acceleration does not by itself mean slowing — it depends on the direction of velocity.

For the common case of uniformly accelerated motion in a straight line (constant $a$), three equations connect the initial velocity $u$, final velocity $v$, acceleration $a$, time $t$ and displacement $s$:

$v = u + at$,   $s = ut + \tfrac{1}{2}at^{2}$,   $v^{2} = u^{2} + 2as$.

These are the workhorses of NEET numericals. Choosing the right one is simply a matter of seeing which quantity is missing: if time is not given, use $v^{2} = u^{2}+2as$; if final velocity is not needed, use $s = ut + \tfrac{1}{2}at^{2}$. A useful fourth relation gives the displacement in the $n$-th second: $s_{n} = u + \tfrac{a}{2}(2n-1)$.

The most-tested special case is free fall under gravity, where $a = g \approx 9.8\ \text{m/s}^{2}$ directed downward. For a body thrown straight up, take a sign convention (say up positive) and apply the same equations with $a = -g$. At the highest point the velocity is momentarily zero but the acceleration is still $g$ downward — a classic NEET trap. The time to rise equals the time to fall back to the same level, and the speed on returning equals the speed of projection.

Motion in a plane needs vectors — quantities with both magnitude and direction, such as displacement, velocity, acceleration and force. They are added not arithmetically but by the triangle or parallelogram law: placing them head-to-tail, the resultant runs from the first tail to the last head. For two vectors $A$ and $B$ at angle $\theta$, the resultant magnitude is $R = \sqrt{A^{2} + B^{2} + 2AB\cos\theta}$, which is largest ($A+B$) when they are parallel and smallest ($|A-B|$) when antiparallel.

The practical tool for NEET is resolution: any vector can be split into perpendicular components $A_x = A\cos\theta$ and $A_y = A\sin\theta$. Working with components turns vector addition into ordinary addition along each axis, which is far less error-prone than geometry. Two perpendicular motions are independent of each other — a key idea that underlies projectile motion.

Relative velocity is how fast one body appears to move as seen from another. The velocity of A relative to B is $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ (vector subtraction). For motion along a line this is just subtraction with signs: two trains moving the same way at $60$ and $40\ \text{km/h}$ have a relative speed of $20\ \text{km/h}$, but moving toward each other the relative speed is $100\ \text{km/h}$.

Two relative-motion situations recur in NEET. A swimmer crossing a river must aim partly upstream so that the resultant of swimming velocity and current carries them straight across; the time to cross depends only on the component of velocity perpendicular to the bank. Rain-and-man problems ask for the apparent direction of rain for a walking observer, found by subtracting the man's velocity from the rain's. In both, drawing the vector triangle is half the solution.

A projectile is any body thrown into the air that then moves only under gravity (air resistance neglected). The single most powerful idea is that horizontal and vertical motions are independent: the horizontal velocity stays constant (no horizontal force), while the vertical motion is uniformly accelerated free fall with $a = g$ downward. Treating the two directions separately, then combining, solves every projectile problem.

For a projectile launched with speed $u$ at angle $\theta$ to the horizontal, the components are $u_x = u\cos\theta$ (constant) and $u_y = u\sin\theta$ (changing under gravity). The path traced is a parabola. At the highest point the vertical velocity is zero but the horizontal velocity $u\cos\theta$ remains, so the speed there is not zero — a frequent NEET misconception.

Three results are worth memorising. The time of flight is $T = \dfrac{2u\sin\theta}{g}$; the maximum height is $H = \dfrac{u^{2}\sin^{2}\theta}{2g}$; and the horizontal range is $R = \dfrac{u^{2}\sin 2\theta}{g}$. From the range formula, the range is maximum at $\theta = 45^{\circ}$, and two complementary angles (such as $30^{\circ}$ and $60^{\circ}$) give the same range — a result NEET tests almost every year.

A simpler special case is horizontal projection from a height $h$ (for example, a ball rolling off a table). Here $u_y = 0$ initially, so the time to fall is $t = \sqrt{2h/g}$, set only by the height, and the horizontal range is $u\cdot t$. The vertical and horizontal parts again do not influence each other: a ball dropped and a ball thrown horizontally from the same height hit the ground at the same instant.