JEE Physics: Kinematics – Full Guide
Kinematics is a fundamental chapter in JEE Physics that deals with the motion of objects without considering the forces causing them. This comprehensive guide covers all essential topics including motion in one and two dimensions, equations of motion, projectile motion, relative velocity, circular motion basics, and includes key formulas, concept explanations, and solved examples to help you master the subject.
1. Introduction to Kinematics
Kinematics describes how objects move — their position, velocity, and acceleration — as functions of time. The study of motion is usually divided into:
- Motion in one dimension (straight line motion)
- Motion in two dimensions (plane motion)
Basic quantities involved:
- Displacement (s): Vector quantity representing the change in position.
- Distance: Scalar quantity representing the total path length travelled.
- Speed: Scalar, rate of change of distance.
- Velocity (v): Vector, rate of change of displacement.
- Acceleration (a): Vector, rate of change of velocity.
2. Motion in One Dimension
2.1 Displacement, Velocity, and Speed
Displacement is given by:
\( \vec{s} = \vec{r}(t) - \vec{r}(t_0) \)
Instantaneous velocity:
\( \vec{v} = \frac{d\vec{s}}{dt} \)
Average velocity:
\( \bar{v} = \frac{\Delta s}{\Delta t} \)
2.2 Equations of Motion for Constant Acceleration
When acceleration \( a \) is constant, the motion is described by the equations:
\[
\begin{cases}
v = u + at \\
s = ut + \frac{1}{2}at^2 \\
v^2 = u^2 + 2as
\end{cases}
\]
Where,
- \( u \): initial velocity
- \( v \): final velocity
- \( a \): acceleration
- \( t \): time
- \( s \): displacement
2.3 Graphical Representation of Motion
Important graphs include displacement vs. time, velocity vs. time, and acceleration vs. time. The slope and area under these graphs provide key insights:
- Slope of displacement-time graph = velocity.
- Area under velocity-time graph = displacement.
- Slope of velocity-time graph = acceleration.
- Area under acceleration-time graph = change in velocity.
3. Motion in Two Dimensions
3.1 Vector Representation of Motion
Motion in a plane is described by vector components along two perpendicular axes, commonly x and y:
\[
\vec{r} = x \hat{i} + y \hat{j}
\]
Velocity and acceleration also have components \( v_x, v_y \) and \( a_x, a_y \).
3.2 Projectile Motion
Projectile motion is a special case of 2D motion under gravity, where the object moves under constant acceleration \( g \) vertically downward, and no acceleration horizontally.
Key formulas assuming launch from ground level with speed \( u \) at angle \( \theta \):
\[
\begin{aligned}
& \text{Horizontal velocity: } v_x = u \cos \theta \\
& \text{Vertical velocity: } v_y = u \sin \theta - g t \\
& \text{Horizontal displacement: } x = u \cos \theta \cdot t \\
& \text{Vertical displacement: } y = u \sin \theta \cdot t - \frac{1}{2} g t^2 \\
& \text{Time of flight: } T = \frac{2 u \sin \theta}{g} \\
& \text{Maximum height: } H = \frac{u^2 \sin^2 \theta}{2g} \\
& \text{Range: } R = \frac{u^2 \sin 2\theta}{g}
\end{aligned}
\]
3.3 Trajectory Equation
Eliminating \( t \) gives the parabolic trajectory:
\[
y = x \tan \theta - \frac{g x^2}{2 u^2 \cos^2 \theta}
\]
4. Relative Velocity
Relative velocity is the velocity of one object as observed from another moving object. If \( \vec{v}_{AB} \) is velocity of A relative to B, and \( \vec{v}_A, \vec{v}_B \) are velocities of A and B relative to ground:
\( \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \)
This concept is crucial for problems involving boats in a stream, trains passing each other, or objects moving on a moving platform.
5. Uniform Circular Motion (Basic Introduction)
An object moving in a circle of radius \( r \) at constant speed \( v \) experiences centripetal acceleration \( a_c \):
\( a_c = \frac{v^2}{r} \)
The velocity vector changes direction continuously though speed is constant.
6. Important Tips and Tricks for Kinematics in JEE
- Always break vectors into components for 2D problems.
- Use equations of motion separately for x and y directions.
- Remember time of flight, maximum height, and range formulas for projectiles.
- Draw neat diagrams labeling all knowns and unknowns.
- Check dimensional consistency of formulas before applying.
- For relative velocity, carefully consider the frame of reference.
7. Solved Example Problems
Example 1: Find the time taken by a car starting from rest and moving with acceleration 2 m/s² to cover 100 m.
Given:
- Initial velocity \( u = 0 \)
- Acceleration \( a = 2 \, m/s^2 \)
- Displacement \( s = 100 \, m \)
Using \( s = ut + \frac{1}{2} a t^2 \), we get:
\( 100 = 0 + \frac{1}{2} \times 2 \times t^2 = t^2 \implies t = 10 \, \text{seconds} \)
Example 2: A projectile is fired at 30° with a velocity of 20 m/s. Find the maximum height reached.
Given:
- Initial speed \( u = 20 \, m/s \)
- Angle \( \theta = 30^\circ \)
- Acceleration due to gravity \( g = 9.8 \, m/s^2 \)
Maximum height formula:
\( H = \frac{u^2 \sin^2 \theta}{2 g} = \frac{(20)^2 \times (\sin 30^\circ)^2}{2 \times 9.8} = \frac{400 \times 0.25}{19.6} = 5.10 \, m \)
8. Summary and Final Notes
Kinematics is the backbone of mechanics and requires a clear understanding of vector quantities, equations of motion, and coordinate decomposition. Make sure to practice diverse problems regularly, focus on conceptual clarity, and memorize key formulas for JEE success.
Consistent practice and solving previous year JEE problems on kinematics can greatly improve your speed and accuracy.