Motion in a Plane — Physics Class 11 Notes (CBSE & HBSE)
Free NCERT Physics notes for Motion in a Plane (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Motion in a Plane (CBSE & HBSE)
Extending kinematics to two dimensions, this chapter develops the mathematics of vectors — their addition, resolution and unit-vector form — and applies it to two important motions: projectile motion (motion under gravity in 2D) and uniform circular motion. Derivations of projectile range, height and time of flight, plus centripetal acceleration, are frequently asked in CBSE and HBSE exams.
Scalars, Vectors, Addition and Resolution
Scalars and Vectors
- A scalar has only magnitude (e.g. mass, time, speed, temperature).
- A vector has both magnitude and direction and obeys the triangle law of addition (e.g. displacement, velocity, force).
Representing and Adding Vectors
Triangle law: If two vectors are represented by the two sides of a triangle taken in order, the resultant is the third side taken in the opposite order.
Parallelogram law: If two vectors A and B act at a point with angle theta between them, the magnitude of the resultant is:
R = sqrt(A^2 + B^2 + 2AB cos theta)
and its direction (angle with A) is tan(alpha) = (B sin theta)/(A + B cos theta).
| Angle theta between vectors | Resultant magnitude | ||
|---|---|---|---|
| 0 degrees | A + B (maximum) | ||
| 90 degrees | sqrt(A^2 + B^2) | ||
| 180 degrees | A - B | (minimum) |
Resolution of a Vector
A vector A making angle theta with the x-axis can be split into rectangular components:
- A_x = A cos theta (horizontal component)
- A_y = A sin theta (vertical component)
Then A = sqrt(A_x^2 + A_y^2) and tan theta = A_y / A_x.
Unit Vectors
A unit vector has magnitude 1 and only indicates direction. The standard unit vectors along x, y, z axes are i, j, k. Any vector is written as:
A = A_x i + A_y j + A_z k
The unit vector along A is A_hat = A / |A|.
CBSE/HBSE trap: Vectors are added by the parallelogram/triangle law, NOT by simple arithmetic — only collinear vectors add or subtract directly.
Projectile Motion
What is a Projectile?
A projectile is any object thrown into the air that moves under gravity alone (air resistance neglected). Its motion is a combination of:
- Horizontal motion with constant velocity (a_x = 0).
- Vertical motion with constant acceleration (a_y = -g).
The two motions are independent; this independence is the key to all projectile problems.
Projectile Launched at Angle theta with Speed u
Initial components: u_x = u cos theta, u_y = u sin theta.
Equation of path (trajectory): eliminating t from x = (u cos theta) t and y = (u sin theta) t - (1/2) g t^2 gives:
y = x tan theta - (g x^2) / (2 u^2 cos^2 theta)
This is the equation of a parabola, so the trajectory is parabolic.
Key Results
| Quantity | Formula |
|---|---|
| Time of flight | T = (2 u sin theta) / g |
| Maximum height | H = (u^2 sin^2 theta) / (2g) |
| Horizontal range | R = (u^2 sin 2theta) / g |
Important Deductions
- Maximum range occurs at theta = 45 degrees, giving R_max = u^2 / g.
- Two angles theta and (90 - theta) give the same range (complementary angles).
- At the highest point, vertical velocity = 0, but horizontal velocity = u cos theta (so speed is not zero there).
Graphs in Words
The path is a symmetric parabola: the projectile rises, reaches maximum height at the midpoint of the range, then falls, hitting the ground at the same speed (magnitude) it was launched.
CBSE/HBSE trap: The horizontal velocity stays constant throughout — gravity has no horizontal component. Only the vertical velocity changes.
Uniform Circular Motion and Centripetal Acceleration
Uniform Circular Motion (UCM)
In uniform circular motion a body moves in a circle of radius r with constant speed, but its velocity changes continuously because its direction changes. Hence the body is accelerating even though speed is constant.
Angular Quantities
- Angular displacement theta (in radians).
- Angular velocity omega = theta / t = d(theta)/dt (unit: rad/s).
- Relation between linear and angular speed: v = r omega.
- Time period T = 2pi / omega; frequency f = 1/T = omega/(2pi).
Centripetal Acceleration
The acceleration in UCM is directed towards the centre of the circle and is called centripetal acceleration:
a_c = v^2 / r = r omega^2 = v omega
The corresponding centripetal force needed is:
F_c = m v^2 / r = m r omega^2
This force is provided by tension, gravity, friction or normal reaction depending on the situation.
Direction Reasoning (Graphs in Words)
As the velocity vector rotates around the circle, the change in velocity delta v always points towards the centre; dividing by delta t shows the acceleration also points to the centre. The velocity is always tangential, while the acceleration is always radial (inward) — the two are perpendicular.
| Quantity in UCM | Direction |
|---|---|
| Velocity v | tangent to circle |
| Centripetal acceleration a_c | towards centre |
| Centripetal force F_c | towards centre |
CBSE/HBSE trap: 'Centrifugal force' is NOT a real force; it is a pseudo-force felt only in a rotating (non-inertial) frame. In an inertial frame, only the inward centripetal force acts.
Frequently asked questions
Are these Motion in a Plane notes free?
Yes — the Motion in a Plane notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Motion in a Plane notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Motion in a Plane chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Motion in a Plane.