Kinematics revision notes

Motion is defined as the change in position of an object with time. The study of motion along a straight line is known as rectilinear motion. In kinematics, we describe motion without concerning ourselves with the causes of the motion. To simplify the analysis, objects in motion are treated as point objects. This approximation is highly valid as long as the size of the object is much smaller than the distance it moves in a reasonable duration of time. In many real-life situations, neglecting the size of objects to treat them as point-like introduces minimal error.

While average velocity describes how fast an object moves over a finite time interval, instantaneous velocity ($v$) defines how fast it moves at a specific instant. It is defined as the limit of the average velocity as the time interval ($\Delta t$) becomes infinitesimally small. Calculus definition: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$. JEE Tip Graphically, the instantaneous velocity at any point is exactly equal to the slope of the tangent to the position-time ($x-t$) curve at that instant.

Speed (instantaneous speed) is strictly defined as the magnitude of instantaneous velocity. For example, velocities of $+24.0 \text{ m s}^{-1}$ and $-24.0 \text{ m s}^{-1}$ both possess an associated speed of $24.0 \text{ m s}^{-1}$. JEE Tip Trap 1 - Average vs Instantaneous Magnitude: A common MCQ trap tests the relationship between speed and velocity. While the average speed over a finite interval is greater than or equal to the magnitude of the average velocity, the instantaneous speed is ALWAYS exactly equal to the magnitude of the instantaneous velocity.

Acceleration represents the rate of change of velocity with time. The average acceleration $\bar{a}$ over a time interval is defined as the change of velocity divided by the time interval: $\frac{v_2 - v_1}{t_2 - t_1}$. Instantaneous acceleration ($a$) is the limit of average acceleration as the time interval goes to zero: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$. Graphically, instantaneous acceleration at a given time is the slope of the tangent to the velocity-time ($v-t$) curve at that instant. Acceleration can result from a change in speed (magnitude of velocity), a change in direction, or both.

Area under the $v-t$ curve: An essential feature of the velocity-time graph is that the area under the curve represents the displacement of the object over a given time interval.

For rectilinear motion with a constant uniform acceleration ($a$), a simple set of kinematic equations governs the relationships between displacement ($x$), time ($t$), initial velocity ($v_0$), and final velocity ($v$). If we assume the particle starts at position $x_0$ at $t = 0$:

1. $v = v_0 + at$
2. $x = x_0 + v_0t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(x - x_0)$

Additionally, the displacement can be expressed using the arithmetic average of the initial and final velocities: $x - x_0 = (\frac{v + v_0}{2})t$.

1. Free Fall: When an object is released near the Earth's surface and falls under the sole influence of gravity (neglecting air resistance), it is in free fall. The acceleration is constant, $a = -g = -9.8 \text{ m s}^{-2}$ (if the upward direction is chosen as positive). The kinematic equations apply exactly, with $v_0 = 0$ if dropped from rest.

2. Galileo’s Law of Odd Numbers: JEE Tip For an object falling from rest under constant acceleration, the distances traversed during equal, successive intervals of time are in the exact ratio of the odd numbers: 1 : 3 : 5 : 7 : 9.... This is an incredible time-saving trick for kinematic ratio problems.

3. Stopping Distance of Vehicles: The distance $d_s$ a vehicle travels before coming to a stop after brakes are applied (causing deceleration $-a$) is given by $d_s = \frac{-v_0^2}{2a}$. JEE Tip Stopping distance is directly proportional to the square of the initial velocity. If a car's initial speed is doubled, its stopping distance increases by a factor of 4.

4. Reaction Time: The time a person takes to observe, think, and act. It can be measured by catching a dropped ruler; using the free-fall equation $d = \frac{1}{2}gt_r^2$, the reaction time is $t_r = \sqrt{\frac{2d}{g}}$.

Introduced to describe motion from different frames, specifically involving 1D moving frames like a police van and a speeding thief's car.

• Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t}$ (Unit: $\text{m s}^{-1}$)
• Instantaneous Velocity: $v = \frac{dx}{dt}$ (Unit: $\text{m s}^{-1}$)
• Average Acceleration: $\bar{a} = \frac{\Delta v}{\Delta t}$ (Unit: $\text{m s}^{-2}$)
• Instantaneous Acceleration (Time Derivative): $a = \frac{dv}{dt}$ (Unit: $\text{m s}^{-2}$)
• Instantaneous Acceleration (Position Derivative): $a = v\frac{dv}{dx}$

• Stopping Distance: $d_s = \frac{v_0^2}{2|a|}$ (Unit: $\text{m}$)
• Reaction Time: $t_r = \sqrt{\frac{2d}{g}}$ (Unit: $\text{s}$)

• Standard Kinematic Equations: The algebraic equations of motion (like $v = v_0 + at$) are strictly valid ONLY for motion where the magnitude and direction of acceleration are completely constant throughout the motion. They cannot be used for variable acceleration.
• Free Fall Approximation: The assumption that $g$ is constant ($9.8 \text{ m s}^{-2}$) is only valid if the height through which the object falls is small compared to the earth's radius. Neglects air resistance.
• Calculus Definitions: The definitions $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt}$ are exact and always correct, regardless of whether acceleration is uniform or non-uniform.
• Point Object: Valid only when object size is much smaller than the distance moved.
• Smoothness of Real-World Graphs: In any realistic situation, acceleration and velocity cannot change values abruptly; thus, graphs for physical motion are always smooth and differentiable without sharp kinks.

Position-Time ($x-t$) Graph

• For uniform motion (zero acceleration), the $x-t$ graph is a straight line inclined to the time axis.
• For constant positive acceleration, the $x-t$ graph is a parabola curving upward.
• For constant negative acceleration, the $x-t$ graph curves downward.
• The slope of the tangent equals instantaneous velocity.

Velocity-Time ($v-t$) Graph

• For uniform motion, it is a horizontal straight line parallel to the time axis.
• For constant acceleration, it is a straight line inclined to the time axis.
• The slope equals instantaneous acceleration.
• The area under the $v-t$ curve equals the exact displacement over that time interval.

JEE Tip Physically Impossible Graphs

• An $x-t$ graph cannot have a vertical line or multiple $x$ values for a single time $t$, as a particle cannot be in multiple places at once.
• A path length (distance) vs. time graph can never have a negative slope because total path length can never decrease over time.
• A speed-time graph can never drop below the x-axis because speed (being a magnitude) cannot be negative.

Deriving Equations of Motion using Calculus

JEE Tip This specific method MUST be used when acceleration is a function of time or position, $a(t)$ or $a(x)$, rather than a constant.

1. Velocity-Time Relation: $a = \frac{dv}{dt} \implies dv = a \cdot dt$. Integrating both sides from $t=0$ to $t=t$ and $v=v_0$ to $v=v$: $\int_{v_0}^{v} dv = \int_{0}^{t} a \cdot dt \implies v = v_0 + at$.
2. Position-Time Relation: $v = \frac{dx}{dt} \implies dx = v \cdot dt$. Substitute $v = v_0 + at$: $dx = (v_0 + at)dt$. Integrating from $x=x_0$ to $x=x$ and $t=0$ to $t=t$: $\int_{x_0}^{x} dx = \int_{0}^{t} (v_0 + at) dt \implies x - x_0 = v_0t + \frac{1}{2}at^2$.
3. Velocity-Position Relation: Using the chain rule: $a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v\frac{dv}{dx}$. Therefore, $v \cdot dv = a \cdot dx$. Integrating from $x=x_0$ to $x=x$ and $v=v_0$ to $v=v$: $\int_{v_0}^{v} v \cdot dv = \int_{x_0}^{x} a \cdot dx \implies [\frac{v^2}{2}]_{v_0}^{v} = a[x]_{x_0}^{x}$. This simplifies to $v^2 - v_0^2 = 2a(x - x_0)$.

• Sign Conventions & Origin Choice: The origin and the positive direction of an axis are entirely a matter of choice. You must specify this choice explicitly before assigning positive or negative signs to displacement, velocity, or acceleration. For example, if upward is chosen as positive, acceleration due to gravity ($g$) is negative ($-9.8 \text{ m s}^{-2}$).

• Zero Velocity $\neq$ Zero Acceleration: JEE Tip The zero velocity of a particle at any given instant does not necessarily imply zero acceleration at that instant. Edge Case: A particle thrown vertically upwards is momentarily at rest ($v=0$) at its uppermost point, but its acceleration continues to be the non-zero acceleration due to gravity.
• Path Length vs. Displacement Magnitude: JEE Tip The magnitude of displacement is strictly equal to the path length if and only if the motion is one-dimensional and strictly in one direction without ever turning back. In all other cases (like a U-turn), path length is strictly greater than the magnitude of displacement.
• Average Speed vs. Magnitude of Average Velocity: Average speed is NOT simply the magnitude of average velocity. If a person walks to a market and walks back home, their overall displacement is zero, making the average velocity zero. However, their average speed is non-zero because the total path length is non-zero.
• Instantaneous Speed vs. Instantaneous Velocity: While average speed and average velocity magnitude can differ, the instantaneous speed is always exactly equal to the magnitude of instantaneous velocity.
• Smoothness of Realistic Graphs: JEE Tip In any physically realistic situation, velocity and acceleration cannot change values abruptly in zero time. Therefore, realistic position-time and velocity-time graphs will be smooth, without any sharp "kinks" (non-differentiable points).
• Kinematic Equations Applicability: The standard kinematic equations are applicable only for one-dimensional motion where both the magnitude and direction of acceleration are completely constant during the course of motion. They completely fail if acceleration varies with time or position.

• Variable Acceleration via Calculus: Extracting equations of velocity or displacement by integrating $a(t)$ or $a(x)$ functions (extensively relying on $a = v \frac{dv}{dx}$).
• Graphical Analysis: Interpreting the areas under $v-t$ graphs for displacement, identifying impossible kinematic graphs via sharp kinks (non-differentiability), and identifying signs of velocity/acceleration from graph concavity.
• Stopping Distance Ratios: Utilizing the proportionality $d_s \propto v_0^2$ to solve collision-avoidance questions efficiently.
• Free Fall & Galileo's Ratio: Using the 1:3:5:7 ratio for rapid calculation of distances fallen in specific seconds of motion.