Kinematics — Revision Notes

⚡ 30-Second Revision

Position: — $vec{r}(t)$ (vector)
Displacement: — $Deltavec{r} = vec{r}_{\text{final}} - vec{r}_{\text{initial}}$ (vector)
Distance: — Total path length (scalar, always $ge 0$)
Average Velocity: — $vec{v}_{\text{avg}} = \frac{Deltavec{r}}{Delta t}$ (vector)
Instantaneous Velocity: — $vec{v} = \frac{dvec{r}}{dt}$ (vector)
Average Acceleration: — $vec{a}_{\text{avg}} = \frac{Deltavec{v}}{Delta t}$ (vector)
Instantaneous Acceleration: — $vec{a} = \frac{dvec{v}}{dt}$ (vector)
**Kinematic Equations (Constant $a$):**

1. $v = u + at$ 2. $s = ut + \frac{1}{2}at^2$ 3. $v^2 = u^2 + 2as$ 4. $s_n = u + \frac{a}{2}(2n - 1)$ (displacement in $n^{\text{th}}$ second)

Projectile Motion (from ground):

- Time of Flight: $T = \frac{2u sin\theta}{g}$ - Maximum Height: $H = \frac{u^2 sin^2\theta}{2g}$ - Horizontal Range: $R = \frac{u^2 sin(2\theta)}{g}$

Relative Velocity: — $vec{v}_{AB} = vec{v}_A - vec{v}_B$

2-Minute Revision

Kinematics describes motion without considering forces. Key quantities are position, displacement (vector, change in position), distance (scalar, total path), velocity (vector, rate of displacement), and acceleration (vector, rate of change of velocity).

For motion with constant acceleration, the three main equations are $v = u + at$, $s = ut + \frac{1}{2}at^2$, and $v^2 = u^2 + 2as$. Remember to use consistent sign conventions for direction. Projectile motion is 2D motion under gravity, where horizontal velocity is constant and vertical motion is uniformly accelerated.

Formulas for time of flight ($T = \frac{2u sin\theta}{g}$), maximum height ($H = \frac{u^2 sin^2\theta}{2g}$), and range ($R = \frac{u^2 sin(2\theta)}{g}$) are essential. Relative velocity ($vec{v}_{AB} = vec{v}_A - vec{v}_B$) is crucial for understanding motion from a moving frame of reference, often requiring vector addition/subtraction.

Graphical analysis (slopes and areas of $x-t$, $v-t$, $a-t$ graphs) is a frequently tested concept.

5-Minute Revision

Kinematics is the language of motion, focusing on 'how' objects move. Start by solidifying the definitions: Position ($vec{r}$) is location, Displacement ($Deltavec{r}$) is the vector change in position, while Distance is the scalar total path length.

Velocity ($vec{v} = dvec{r}/dt$) is the rate of displacement, and Acceleration ($vec{a} = dvec{v}/dt$) is the rate of change of velocity. Remember, velocity and acceleration are vectors, meaning direction matters.

An object can have constant speed but changing velocity (e.g., uniform circular motion) if its direction changes, implying acceleration.

For constant acceleration (1D motion), the three core equations are your best friends:

1
$v = u + at$ (final velocity = initial velocity + acceleration $imes$ time)
2
$s = ut + \frac{1}{2}at^2$ (displacement = initial velocity $imes$ time + half $imes$ acceleration $imes$ time squared)
3
$v^2 = u^2 + 2as$ (final velocity squared = initial velocity squared + 2 $imes$ acceleration $imes$ displacement)

Also, know the displacement in the $n^{\text{th}}$ second: $s_n = u + \frac{a}{2}(2n - 1)$. Always choose a consistent positive direction.

Projectile motion is 2D motion under constant gravity. The key insight is to treat horizontal and vertical motions independently. Horizontal motion has constant velocity ($u_x = u cos\theta$, $a_x = 0$), while vertical motion is uniformly accelerated ($u_y = u sin\theta$, $a_y = -g$).

Important formulas are: Time of Flight $T = \frac{2u sin\theta}{g}$, Maximum Height $H = \frac{u^2 sin^2\theta}{2g}$, and Horizontal Range $R = \frac{u^2 sin(2\theta)}{g}$. Remember that range is maximum at $45^circ$ and equal for complementary angles.

Relative motion involves understanding motion from a moving frame. The core formula is $vec{v}_{AB} = vec{v}_A - vec{v}_B$. This is crucial for river-boat problems (e.g., shortest time vs. shortest path) and rain-man problems. Always use vector subtraction carefully.

Graphical analysis is vital. The slope of an $x-t$ graph gives velocity; the slope of a $v-t$ graph gives acceleration. The area under a $v-t$ graph gives displacement; the area under an $a-t$ graph gives change in velocity. Practice interpreting curves for non-uniform motion.

Example: A stone is dropped from a tower of height $45,\text{m}$. How long does it take to hit the ground? (Take $g=10,\text{m/s}^2$)

Given: $u=0$ (dropped), $s=45,\text{m}$ (taking downward as positive), $a=g=10,\text{m/s}^2$.
Formula: $s = ut + \frac{1}{2}at^2$
Calculation: $45 = 0 \times t + \frac{1}{2}(10)t^2 Rightarrow 45 = 5t^2 Rightarrow t^2 = 9 Rightarrow t = 3,\text{s}$.

Prelims Revision Notes

Kinematics: NEET UG Revision Notes

1. Fundamental Quantities & Definitions:

Position ($vec{r}$): — Vector from origin to object. $xhat{i} + yhat{j} + zhat{k}$.
Distance (scalar): — Total path length. Always $ge 0$.
Displacement ($Deltavec{r}$): — Vector change in position. $vec{r}_{\text{final}} - vec{r}_{\text{initial}}$. Can be positive, negative, or zero.
Speed (scalar): — Rate of distance covered. $|vec{v}|$.
Velocity ($vec{v}$ vector): — Rate of displacement. $vec{v} = \frac{dvec{r}}{dt}$. Direction matters.

* Average velocity: $rac{Deltavec{r}}{Delta t}$. * Instantaneous velocity: $lim_{Delta t \to 0} \frac{Deltavec{r}}{Delta t}$.

Acceleration ($vec{a}$ vector): — Rate of change of velocity. $vec{a} = \frac{dvec{v}}{dt}$.

* Average acceleration: $rac{Deltavec{v}}{Delta t}$. * Instantaneous acceleration: $lim_{Delta t \to 0} \frac{Deltavec{v}}{Delta t}$.

2. Motion with Constant Acceleration (1D):

Equations of Motion:

1. $v = u + at$ 2. $s = ut + \frac{1}{2}at^2$ 3. $v^2 = u^2 + 2as$ 4. Displacement in $n^{\text{th}}$ second: $s_n = u + \frac{a}{2}(2n - 1)$

Free Fall: — Special case where $a = pm g$ (acceleration due to gravity, $approx 9.8,\text{m/s}^2$ or $10,\text{m/s}^2$). Choose consistent sign convention (e.g., upward positive, downward negative).

3. Projectile Motion (2D):

Assumptions: — Negligible air resistance, constant $g$ downwards.
Independent Motions:

* Horizontal: Uniform velocity ($a_x = 0$). $x = (u cos\theta)t$, $v_x = u cos\theta$. * Vertical: Uniform acceleration ($a_y = -g$). $y = (u sin\theta)t - \frac{1}{2}gt^2$, $v_y = u sin\theta - gt$, $v_y^2 = (u sin\theta)^2 - 2gy$.

Key Formulas (from ground):

* Time of Flight ($T$): $T = \frac{2u sin\theta}{g}$ * Maximum Height ($H$): $H = \frac{u^2 sin^2\theta}{2g}$ * Horizontal Range ($R$): $R = \frac{u^2 sin(2\theta)}{g}$ * Max Range at $heta = 45^circ$. Ranges are equal for $heta$ and $(90^circ - \theta)$.

4. Relative Motion:

Relative Velocity: — $vec{v}_{AB} = vec{v}_A - vec{v}_B$ (velocity of A with respect to B).
River-Boat Problems:

* Shortest time to cross: Boat heads perpendicular to river flow. $t = \frac{W}{v_B}$, drift $x = v_R t$. * Shortest path (no drift): Boat heads upstream at an angle. $v_B sin\theta = v_R$, $t = \frac{W}{v_B cos\theta}$.

Rain-Man Problems: — $vec{v}_{RM} = vec{v}_R - vec{v}_M$. Angle of umbrella depends on $vec{v}_{RM}$.

5. Graphical Analysis:

**Position-Time ($x-t$) Graph:**

* Slope = Velocity. * Straight line: Constant velocity. * Curve: Changing velocity (acceleration).

**Velocity-Time ($v-t$) Graph:**

* Slope = Acceleration. * Area under curve = Displacement. * Straight line: Constant acceleration. * Horizontal line: Constant velocity (zero acceleration).

**Acceleration-Time ($a-t$) Graph:**

* Area under curve = Change in velocity.

Common Traps:

Confusing distance/displacement, speed/velocity.
Incorrect sign conventions for vectors.
Ignoring vector nature in 2D/relative motion.
Misinterpreting graph slopes/areas.

Vyyuha Quick Recall

SUVAT for Kinematic Equations:

S = Displacement U = Initial Velocity V = Final Velocity A = Acceleration T = Time

Think of a 'SUV AT' the starting line, ready to accelerate! This helps remember the variables involved in the equations of motion.