Motion in a Plane

8 MCQs3 revision cards9-step worked example
Source: NCERT KinematicsPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

Motion in a plane combines two independent one-dimensional motions — horizontal and vertical — into a single framework. The core insight: a projectile's horizontal velocity stays constant (no horizontal force, air resistance neglected), while its vertical velocity changes under gravity alone (NCERT Class 11 Physics Chapter 3, page 10). This decomposition is the engine behind every projectile and circular-motion problem NEET asks.

Projectile traps that cost marks.

The height formula H = v₀² sin²θ / (2g) and the range formula R = v₀² sin(2θ) / g share v₀ and θ but differ in the sin-vs-sin² factor and the denominator. A common confusion: plugging sin(2θ) when the question asks for height, or sin²θ when it asks for range. At θ = 45°, both formulas give the same numerator — that's the checkpoint to verify you picked the right one.

At the highest point of a projectile's trajectory, the vertical component vanishes. The speed there equals v₀ cosθ — not zero, not v₀ sinθ. Watch the angle reference: "60° with the vertical" means 30° with the horizontal. The horizontal component switches from cosθ to sinθ when the reference axis flips.

Uniform circular motion (UCM) — the velocity-vs-speed distinction.

In UCM, speed is constant but velocity is not. Direction changes continuously, producing centripetal acceleration a_c = v²/r directed inward. A question asking "which quantity is constant?" tests whether you treat velocity as a scalar (wrong) or a vector (correct). Speed and kinetic energy are constant; velocity and acceleration direction are not.

The UCM-to-projectile bridge.

A particle completing circular motion at radius R with period T has speed v = 2πR/T. If that particle is then launched vertically, the max height is H = v²/(2g) = (2πR/T)²/(2g). The trap: using R directly as the height.

Relative velocity in 2D is vector subtraction — same-direction objects have relative velocity v_A − v_B (smaller magnitude); opposite-direction objects have v_A + v_B (larger magnitude). Sign errors here are common under time pressure.


Practice MCQs

Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.

MCQ 1Direct ApplicationPYQ Pattern

A projectile is launched with speed 20 m/s at 30° above the horizontal. What is the maximum height reached? (Take g = 10 m/s²)

MCQ 2Concept TrapPYQ Pattern

A projectile is launched at 60° with the vertical. At the highest point, the speed of the projectile is v₀/2. What is the launch angle with respect to the horizontal?

MCQ 3Easy RecallPYQ Pattern

In uniform circular motion at constant speed, which of the following is constant?

MCQ 4Direct ApplicationPractice

A particle moves in a circle of radius 0.50 m with a constant speed of 2.0 m/s. The magnitude of centripetal acceleration is:

MCQ 5Easy RecallPractice

A body moving in a circle at constant speed has:

MCQ 6Easy RecallPractice

A projectile is launched at speed v₀ at angle θ above the horizontal. Which expression gives the horizontal range on level ground?

MCQ 7CalculationPYQ Pattern

A particle completes uniform circular motion with radius R = 0.10 m and period T = 0.20 s. It is then launched vertically upward with the same speed. The maximum height reached is: (Take g = 10 m/s², π² ≈ 10)

MCQ 8CalculationPYQ Pattern

Two buses travel a route of length L. Bus A and Bus B move in opposite directions, each at speed v_b. A car travels the route at speed v_c in the same direction as Bus A. If buses from each end depart every T minutes, the car encounters an oncoming bus (Bus B) every T_opp = L/(v_b + v_c) minutes and a same-direction bus (Bus A) overtakes or is overtaken every T_same = L/(v_b − v_c) minutes. If T_opp = 5 min and T_same = 15 min, the ratio v_b/v_c is:

Quick recall before you leave

Worked Example

Pattern: Projectile launched from UCM — find maximum height (pattern: projectile from circular motion).

  1. 1

    Given

    A particle moves in uniform circular motion with radius R = 0.20 m and period T = 0.40 s. It is then launched vertically upward with the same speed it had in circular motion. Take g = 9.8 m/s², π² = 9.87.

  2. 2

    Required

    Maximum height H reached by the particle after vertical launch.

  3. 3

    Concept

    The particle's speed in UCM is v = 2πR/T. Once launched vertically, it becomes a projectile with initial upward speed v and deceleration g. At the peak, all kinetic energy converts to potential energy: H = v²/(2g).

  4. 4

    Formula

    v = 2πR/T H = v²/(2g)

  5. 5

    Substitution

    v = 2π(0.20)/0.40 = π m/s H = (π)²/(2 × 9.8) = π²/19.6

  6. 6

    Calculation

    H = 9.87/19.6 = 0.5036 m Note on exact constants: the factor 2 in the denominator and the formula structure (2πR/T) are exact mathematical relations. Only R, T, and g carry measurement precision. With R (2 sig figs), T (2 sig figs), and g (2 sig figs from 9.8), the result is limited to 2 significant figures.

  7. 7

    Final answer

    H ≈ 0.50 m (2 significant figures)

  8. 8

    Common trap

    Using R = 0.20 m directly as the height (the UCM-to-projectile bridge trap). The radius has no direct geometric relationship to the projectile's maximum height — the speed must be computed first from v = 2πR/T.

  9. 9

    Similar NEET-style question

    A satellite moves in a circular orbit of radius 6400 km with period 90 minutes. If an object at the same speed were launched vertically from the surface, what maximum height would it reach? (Same two-step pattern: compute v from UCM, then apply H = v²/(2g).) ---

Before solving, remember these

In two-dimensional motion, the components along perpendicular axes evolve independently under the corresponding components of acceleration. This decoupling is the key insight that reduces 2D motion to two 1D problems.

-- NCERT Class 11 Physics, Ch. 3, p. 10

Formulas

6 formulas — click to collapse

First kinematic equation (uniform acceleration)

Final velocity equals initial velocity plus acceleration times the time elapsed, for motion under constant acceleration.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant (uniform) accelerationm/s^2
tElapsed times

Valid when

  • Acceleration a is CONSTANT (uniform) in both magnitude and direction
  • All quantities measured in the same inertial reference frame
  • Motion is along a straight line; signs encode direction along chosen axis

Do NOT use when

  • Acceleration changes in magnitude or direction (use a(t) integration)
  • Motion is uniformly circular at constant speed (a is centripetal, not tangential)

Second kinematic equation (displacement under uniform acceleration)

Displacement equals initial-velocity-times-time plus half of acceleration-times-time-squared. The (1/2) factor is the area of the triangle on the v-t graph.

SymbolQuantitySI Unit
xFinal positionm
x0Initial positionm
v0Initial velocitym/s
aConstant accelerationm/s^2
tTime elapseds

Valid when

  • Acceleration constant (magnitude and direction)
  • Sign convention consistent across x, v, a (one chosen positive direction)

Third kinematic equation (velocity-squared)

Relates final velocity to initial velocity, displacement, and acceleration without using time. Most useful when t is unknown or unwanted.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant accelerationm/s^2
x - x0Displacementm

Valid when

  • Constant acceleration
  • Use signed values for v, v0, a, and (x - x0) consistently

Do NOT use when

  • Time-dependent acceleration
  • Curvilinear motion where acceleration is not parallel to displacement

Projectile maximum height

Maximum height attained by a projectile launched at speed v0 and angle theta0 above the horizontal, measured above the launch level.

SymbolQuantitySI Unit
HMaximum height (above launch)m
v0Launch speedm/s
theta0Launch anglerad/deg
gGravitational accelerationm/s^2

Valid when

  • Air resistance neglected
  • Constant g over trajectory

Projectile horizontal range

For a projectile launched from and returning to the same horizontal level with initial speed v0 at angle theta0 above the horizontal, the horizontal range R is given by this formula. R is maximised at theta0 = 45 deg.

SymbolQuantitySI Unit
RHorizontal rangem
v0Launch speedm/s
theta0Launch angle above horizontalrad (or deg with sin in deg)
gGravitational accelerationm/s^2

Valid when

  • Launch and landing are at the same vertical height
  • Air resistance neglected
  • g treated as constant over the trajectory

Do NOT use when

  • Launch and landing heights differ (use full kinematics)
  • Significant air drag (e.g. table-tennis ball, badminton shuttle)
  • Variation of g (ballistic trajectories spanning large altitude changes)

Centripetal acceleration in uniform circular motion

An object moving in a circle of radius r at constant speed v has acceleration of magnitude v^2/r (or equivalently omega^2 * r) directed toward the centre. This is centripetal (radially inward), not tangential.

SymbolQuantitySI Unit
a_cCentripetal accelerationm/s^2
vTangential speedm/s
rRadius of circlem
omegaAngular speedrad/s

Valid when

  • Speed v is constant (uniform circular motion)
  • r and the centre are well-defined (instantaneous radius of curvature for general curved motion)

Do NOT use when

  • Non-uniform circular motion (then there is also a tangential acceleration component)

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

14 items — click to collapse

Category: Similar Terms

Student answers 1:2:3:4 for distances in successive 1-second intervals (linear) instead of 1:3:5:7 (Galileo's odd numbers).

When it triggers

Question asks about ratios of distances traversed in successive 1-s intervals during free fall from rest.

How to avoid

Distance grows quadratically (y = ½ g t²); successive interval distances are y_n - y_{n-1} = ½ g (t_n² - t_{n-1}²) = ½ g (2n-1) seconds. The factor (2n-1) gives 1, 3, 5, 7, ...

Category: Graph Interpretation

Student uses sin or cos of the angle the line makes with the time axis, instead of tan, to extract velocity.

When it triggers

Question gives an angle the x-t line makes with the t-axis (often 30°, 45°, 60°) and asks for velocity or its ratio.

How to avoid

Velocity = dx/dt = slope of x-t line = tan(angle), where the angle is measured from the time axis. Always tan, not sin or cos.

Category: Overthinking

Student attempts to invert t(x) algebraically before differentiating, getting tangled in messy algebra; misses chain rule.

When it triggers

Question gives t as function of x (instead of x as function of t), e.g. t = x² + x.

How to avoid

Differentiate the given relation directly: dt/dx = (function of x). Then v = dx/dt = 1/(dt/dx). For acceleration use chain rule: a = dv/dt = (dv/dx)(dx/dt) = v dv/dx.

Category: Sign Convention

Student treats a 'thrown vertically downward' problem as if the object were dropped (u = 0). The result is wrong by an additive u² term in v² = u² + 2gh. When the question explicitly states a launch speed, that speed is non-zero and CANNOT be ignored.

When it triggers

Question phrases: 'thrown vertically downward', 'projected with initial velocity', 'launched with speed u'. If u is given numerically, it MUST appear in the equation.

How to avoid

Always parse the launch verbal cue and write down u with its sign before reaching for v² = 2gh. Use the full v² = u² + 2gh (or u² - 2gh for upward motion).

Category: Similar Terms

Student plugs into v₀² sin(2θ)/g (range) when asked for maximum height, or vice versa. The two share v₀ and θ but have different sin-vs-sin² and 2g-vs-g terms.

When it triggers

Question mentions launch speed and angle and asks for max height (H) or range (R). Distractors include the wrong formula's answer.

How to avoid

Memorise BOTH formulas explicitly: H = v₀² sin² θ / (2g) (note sin²); R = v₀² sin(2θ) / g (note sin of doubled angle). Check by setting θ = 45°: max range, half max height.

Category: Sign Convention

Student plugs angle θ into v cos θ when the question states 'angle with the vertical' (which makes the horizontal component v sin θ).

When it triggers

Question phrases like 'thrown at angle θ with the vertical direction' or 'with horizontal'.

How to avoid

Always identify reference axis explicitly. From horizontal: vx = v cos θ, vy = v sin θ. From vertical: vx = v sin θ, vy = v cos θ. The two are complementary (θ_h + θ_v = 90°).

Category: Sign Convention

Student fails to distinguish between same-direction and opposite-direction relative velocities, treating both as magnitudes.

When it triggers

Question describes two objects moving on the same line; observer somewhere between or alongside.

How to avoid

Relative velocity is a VECTOR. Same direction: v_rel = v_a - v_b (smaller magnitude). Opposite direction: v_rel = v_a + v_b (larger magnitude). Use sign convention consistently along chosen axis.

Category: Overthinking

Student assumes proportionality of speed to remaining distance under uniform deceleration. In fact, KE drops linearly with distance (v² is the linear quantity, not v): v² = u² - 2as. Speed-vs-distance is a sqrt-curve, not a line.

When it triggers

Question describes a body decelerating through stages with given speed at one stage; asks for distance to stop or speed at another stage.

How to avoid

Always work with v², not v, when uniform deceleration is in play. The work-energy theorem gives the same answer faster: ½ m v² = work done against constant force over distance.

Category: Overthinking

Student uses the radius R as the projectile launch height or fails to compute the UCM speed from period.

When it triggers

Question describes a particle in UCM with given (R, T) then says 'now launched vertically up with same speed; find max height'.

How to avoid

Step 1: speed v = 2πR/T (from UCM). Step 2: max projectile height H = v²/(2g) = (2πR/T)² / (2g). Don't shortcut by setting H = R.

Category: Similar Terms

Student claims velocity is constant in uniform circular motion (it's not — direction changes).

When it triggers

Question asks 'in uniform circular motion at constant speed, which is also constant?'

How to avoid

In UCM: SPEED constant; KE constant. VELOCITY (vector) NOT constant. ACCELERATION (centripetal, magnitude v²/r) constant in MAGNITUDE but NOT in direction.

Root cause: formula misuse

Correction

The three kinematic equations require CONSTANT acceleration. For variable acceleration, use a = dv/dt and integrate, or use v dv = a dx for position-dependent acceleration. Verify constant-a before applying these formulas.

Wrong option pattern

Distractor uses constant-acceleration kinematic equations on a problem where the question explicitly says acceleration changes with time or position.

Root cause: concept gap

Correction

Standard range R = v0^2 sin(2*theta)/g and H = v0^2 sin^2(theta)/(2g) assume (i) launch and landing at the same height, (ii) negligible air drag, and (iii) constant g. For asymmetric trajectories, use the full kinematic decomposition along x and y.

Wrong option pattern

Distractor applies R = v0^2 sin(2*theta)/g to a projectile launched from a cliff.

Past Year Questions

10 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
NEET 2025

Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

115 min, 120 km/h
29 min, 40 km/h
325 min, 100 km/h
410 min, 90 km/h
NTA Answer: Option 1(final)
NEET 2021

A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution. If this particle were projected with the 'θ' same speed at an angle to the horizontal, the maximum height attained by it equals 4R. The θ, angle of projection, is then given by : 1 2gT 2 2 θ=sin−1 

1π2  R  1  2  2 gT θ=cos−1
2  π2 R 1 π2  2 R θ=cos−1 
32 gT  1 π2  2 R θ=sin−1 
42 gT 
NTA Answer: Option 1(final)

How NEET usually asks this

10 recurring patterns from past papers — click to collapse

A projectile (typically a bullet) penetrates a uniform medium with constant decelerating force; given initial speed and speed after a known distance, find the total stopping distance. Apply v² = u² + 2as to each segment, noting that 'a' is the same throughout. Common shape: bullet hits block at u, slows to u/k after distance d₁; how much further to stop?

Multi StepMedium

Common distractors

treats speed ratio as distance ratio

Linear thinking: 'speed went from u to u/3, so distance traversed should be 3× the original'

Object given a non-zero downward initial velocity from elevation; asked for the height fallen, time of impact, or final speed. Apply v² = u² + 2gh (or analogous) with proper signs. Common shape: a ball thrown vertically downward from a tower with initial speed u, hitting the ground at speed v; find the tower height. Distractors test (i) sign-of-u confusion, (ii) using v² = 2gh forgetting u², (iii) wrong g unit.

Multi StepEasy

Common distractors

drops initial velocity term

Student conflates 'thrown' with 'dropped' and uses v² = 2gh

Sources

NCERT refs: Class 11 Physics Chapter 3, p.10

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