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Motion in a Plane
Physics · Class 11NCERT pp. 27–432 sets × 25 questionsFree
Key facts and formulas for Motion in a Plane, three worked sample questions, and two full practice sets. Every question has a hint, a reason for every option and a step-by-step solution, with a calculator or animation where one helps. English and Hinglish.
Formula sheet for this chapter
| Idea | Rule or fact |
|---|
| Resultant of two vectors | R = √(P² + Q² + 2PQ cosθ); θ = 0° gives P + Q, 180° gives |P − Q|, 90° gives √(P² + Q²) |
| Components | Ax = A cosθ, Ay = A sinθ; magnitude √(Ax² + Ay²); tanθ = Ay/Ax (check the quadrant) |
| Unit vector and dot product | Â = A/|A|; A·B = AxBx + AyBy = AB cosθ; A·B = 0 means perpendicular |
| Velocity and acceleration | v_avg = Δr/Δt, v = dr/dt, a_avg = Δv/Δt, a = dv/dt; motions along x and y are independent |
| Projectile at angle θ | T = 2u sinθ/g, H = u² sin²θ/2g, R = u² sin2θ/g; R/H = 4/tanθ |
| Projectile facts | vx is constant; at the top vy = 0 and speed = u cosθ; R is maximum at 45° (u²/g); same R for θ and 90° − θ |
| Horizontal projectile | t = √(2h/g), range = v t, landing angle tanφ = gt/vx |
| Relative velocity | v_AB = v_A − v_B; river crossing time = width/(speed across); straight across needs u sinα = current |
| Circular motion | ω = 2π/T = 2πν, v = ωR, T = 2πR/v; a = v²/R = ω²R toward the centre |
| Uniform circular motion | Speed is constant but velocity changes; a is perpendicular to v and its direction keeps rotating; |Δv| for a quarter turn = √2 v |
Three sample questions
Q1. Which pair contains only vector quantities?
- speed and distance
- displacement and velocity
- mass and force
- temperature and acceleration
Hint and solution
Hint: A vector needs both magnitude and direction. Ask for each quantity: does it make sense to give a direction?
- Vector = magnitude + direction — A vector is a quantity that needs both a size and a direction to be fully described.
- Scalars: speed, distance, mass, temperature — These four only have size. So any pair containing them cannot be all vectors.
- Vectors: displacement, velocity, force, acceleration — Eliminate pairs with speed, distance, mass or temperature. Only displacement and velocity remain together.
- Answer: displacement and velocity — Both are vectors, so the answer is Option B.
Answer: B. Displacement and velocity both need a direction. Speed, distance, mass and temperature are scalars.
Q2. Statement I: Scalars can be added only if they have the same unit. Statement II: Vector addition is commutative.
- Statement I is true, Statement II is false
- Statement I is false, Statement II is true
- Both statements are false
- Both statements are true
Hint and solution
Hint: Can you add 3 kg to 2 s? And does A + B give the same vector as B + A?
- Check Statement I — Adding scalars of different units, like mass and time, has no meaning. So I is true.
- Check Statement II: A + B = B + A — In the triangle or parallelogram method the resultant is the same whichever vector is drawn first. So II is true.
- Both true — Both statements are correct standard results.
- Answer: both true — So the correct choice is Option D.
Answer: D. Both are standard results from this chapter.
Q3. Two perpendicular vectors have magnitudes 8 and 6. The magnitude of their resultant is
- 10
- 14
- 2
- 7
Hint and solution
Hint: At 90° the two vectors form the sides of a right triangle. The resultant is the hypotenuse.
- R = √(P² + Q²) at θ = 90° — For perpendicular vectors the cosine term in the general formula becomes zero.
- R² = 8² + 6² = 64 + 36 = 100 — Square each magnitude and add them.
- R = √100 = 10 — Take the square root.
- Resultant = 10 — The magnitude is 10 units, which is Option A.
Answer: A. R = √(8² + 6²) = 10.
How to use this chapter
Read the NCERT pages first, then attempt Set 1 without the hints. Note every wrong answer in a mistake list, read the reason for each option, and retry Set 2 two days later. A topic is solid when you can say why each wrong option is wrong.