Chapter 3: Motion in a Plane
Original notes — ready CBSE (NCERT, English Medium) • Class 11 • Physics • Book: Physics — Part I
Motion in a Plane — EG's own summary, in simple words
Book: Physics — Part I (CBSE, Class 11) | Chapter 3 | Medium: English
Most real motion is not along one line — a ball thrown in the air, a car turning a corner, the Moon around the Earth. Motion in a plane needs two directions, and the tool that handles direction is the vector. This chapter teaches you to split motion into parts and to put them back together.
Diagram (labels on the image): A ball above a field with a dotted path rising from the ground. The dotted line marks the ball's flight path — note honestly, it is drawn wavy here, while a real projectile path is a smooth parabola. Horizontal velocity stays constant while gravity pulls the ball down vertically.
1. Scalars and vectors
A scalar has only magnitude: mass, temperature, distance, speed. A vector has magnitude and direction: displacement, velocity, acceleration, force. The same number can mean different things as a vector — 5 m east is not the same as 5 m north. Vectors are drawn as arrows: the length shows magnitude and the arrowhead shows direction. Adding vectors is not like adding numbers: 3 m east + 4 m north does not give 7 m, it gives 5 m north-east, by the triangle rule.
2. Adding and resolving vectors
Two vectors are added by placing them head to tail; the result, the resultant, runs from the start of the first to the end of the second. The parallelogram rule gives the same result. The reverse process is resolution: a vector is split into components along two perpendicular directions, usually x and y. If a vector of magnitude A makes an angle θ with the x-axis, its x-component is A cos θ and its y-component is A sin θ. Resolution is the key to almost every problem in this chapter — split first, solve each direction separately, then combine.
3. Projectile motion
A projectile is an object thrown into the air that moves under gravity alone, like a cricket ball or a stone. Its motion splits neatly: horizontally, there is no acceleration, so horizontal velocity stays constant; vertically, gravity accelerates it downwards at g. The path that results is a parabola. Time of flight depends only on the vertical motion, range depends on both, and the maximum height depends on the vertical part of the initial velocity. The greatest range for a given speed comes at a 45° launch angle.
4. Uniform circular motion
An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. This centripetal acceleration points towards the centre of the circle and has magnitude v²/r, where v is speed and r is radius. The time for one full round is the time period, T = 2πr/v. A satellite in orbit, a car on a roundabout and a stone whirled on a string are all examples: something must always pull towards the centre, or the object would fly off in a straight line.
5. Relative velocity — a useful idea
Velocity depends on who is watching. A person walking inside a moving train moves slowly relative to the train but fast relative to the ground. Relative velocity of A with respect to B is the velocity of A minus the velocity of B, done as vectors. Rain appearing to fall slanting to a moving cyclist, and a boat crossing a river being carried downstream, are the two classic problems — both are solved by drawing the vector triangle carefully.
Chart / Table — Projectile motion at a glance (launch speed u, angle θ)
| Quantity | Formula | Depends on |
|---|---|---|
| Time of flight | T = 2u sin θ / g | vertical velocity only |
| Maximum height | H = u² sin² θ / 2g | vertical velocity only |
| Horizontal range | R = u² sin 2θ / g | both components; max at 45° |
Caption: Every kick is a projectile problem: the launch angle and speed decide how high the ball goes and how far it lands.
Practice set — try it yourself
- A vector of magnitude 10 units makes 30° with the x-axis. Find its x and y components. Hint: x = A cos θ, y = A sin θ.
- A ball is thrown at 20 m/s at 30° above the horizontal. Find its horizontal and vertical velocity components. Hint: Resolve the 20 m/s into cos and sin parts.
- Why does a projectile's horizontal velocity stay constant while its vertical velocity changes? Hint: Ask what force acts in each direction.
- A stone is whirled in a circle of radius 0.5 m at 4 m/s. Find its centripetal acceleration. Hint: Use v²/r, pointing to the centre.
- A boat tries to cross a river straight across but lands downstream. Explain using relative velocity. Hint: Add the boat's velocity and the river's velocity as vectors.
Key Terms / Important Terms
- Scalar — a quantity with magnitude only, like mass or temperature.
- Vector — a quantity with magnitude and direction, like velocity or force.
- Resultant — the single vector that has the same effect as two or more vectors added together.
- Resolution — splitting a vector into components along chosen directions.
- Projectile — an object moving under gravity alone after being thrown.
- Trajectory — the path followed by a projectile — a parabola.
- Time of flight — the total time a projectile stays in the air.
- Range — the horizontal distance a projectile covers.
- Centripetal acceleration — acceleration towards the centre of a circular path, v²/r.
- Relative velocity — velocity of one object as seen from another moving object.
Remember
- Vector addition is head-to-tail, not simple number addition.
- Resolve first: x-component A cos θ, y-component A sin θ.
- In projectile motion, horizontal velocity is constant; vertical motion has acceleration g downwards.
- Maximum range comes at 45°; time of flight depends only on vertical motion.
- Circular motion at constant speed still has acceleration — towards the centre.
Exam Tip / Common Mistake
Common mistake: using the full launch speed where only a component belongs. Range and height problems fail when students put u instead of u cos θ (horizontal) or u sin θ (vertical). Always resolve the velocity first, and label which component you are using in each step.
Questions with Answers — full answers
- Why is displacement a vector but distance a scalar? Answer: Displacement needs a direction to be complete — 5 m east and 5 m north are different displacements. Distance only needs a size, so it is a scalar.
- A projectile is launched at 45°. Why does it have the maximum range at this angle? Answer: Range R = u² sin 2θ / g. Sin 2θ is largest (equal to 1) when 2θ = 90°, that is, θ = 45°, so the range is greatest there.
- A car goes round a bend at constant speed. Is it accelerating? Explain. Answer: Yes. Its direction changes continuously, and acceleration is a change in velocity, which includes direction. The acceleration points towards the centre of the bend.
- A ball is thrown horizontally from a height and another is dropped from the same height at the same moment. Which lands first, and why? Answer: Both land together. Vertical motion is the same for both (initial vertical velocity zero, acceleration g), and horizontal motion does not affect the time to fall.
- Resolve a force of 50 N acting at 60° to the horizontal into horizontal and vertical components. Answer: Horizontal = 50 cos 60° = 25 N. Vertical = 50 sin 60° ≈ 43.3 N.