PYQ Vault

Playbook

Motion in a Plane

Projectiles, relative velocity and circular motion. Lighter than in the early papers, but each question is a vector split into two perpendicular parts.

Questions in the bank
151
q/paper in 2025–26
0.82
Numeric answer
24%
Notes pages
6

Tier: Long tail

When you’ll see it

Two vectors to add or resolve, a projectile, a boat crossing a river, rain seen from a moving body, or a body going round a circle.

How this chapter is tested

Almost every question splits a vector into two perpendicular parts: x and y for a projectile, across and along for a river, towards the centre and along the path for a circle. Once the split is right, each direction is a one-dimensional problem.

Projectiles and circular motion carry most of the chapter. The circular questions lean on forces as much as on kinematics, because friction, a string, a spring or a wall has to supply mv²/r.

The chapter is set less often than in the early papers. Marks are lost on sin 2θ written for sin²θ, on an angle quoted against the wrong axis, and on a speed taken as zero at the top of a path where it is not.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Vectors: resultant, components and products

    R² = A² + B² + 2AB cos θ; resolve along x and y; A·B = AB cos θ tests perpendicularity, and A × B is at right angles to both.

  • Velocity in a plane and relative velocity

    Differentiate the position one component at a time; the velocity of A seen from B is v_A − v_B, which settles rivers and rain.

  • Time of flight, height and range

    T = 2u sin θ/g, H = u² sin²θ/2g, R = u² sin 2θ/g; the range is greatest at 45°, and θ and 90° − θ give the same range.

  • Velocity, trajectory and launch from a height

    u cos θ never changes; y = x tan θ − gx²/(2u² cos²θ); a body thrown horizontally from h falls for √(2h/g).

  • Uniform circular motion

    v = ωr and a = v²/r towards the centre even at constant speed; a tangential acceleration adds at right angles.

  • Roads, strings and vertical circles

    Name the force that supplies mv²/r: friction, banking, a string, a spring or a wall; in a vertical circle the speed changes from top to bottom.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • sin 2θ and sin²θ swapped

    The range has sin 2θ and the height sin²θ. Check with 90°, where the range must be zero and the height u²/2g.

  • An angle against the wrong axis

    A velocity 4i − j m/s is at tan⁻¹(1/4) below the +x axis. Options give the right number against the wrong axis.

  • Zero speed at the top

    At the top of a projectile only the vertical velocity is zero; the speed is u cos θ. On a string, just completing a vertical circle needs √(gr) at the top.

  • Accelerations added directly

    Tangential and centripetal accelerations are at right angles, so the total is √(a_t² + a_c²), never a_t + a_c.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Motion in a Plane notes

Drill every Motion in a Plane question

151 questions from the bank, across 6 subtopics.

Drill the 151 questions

Drill one subtopic at a time

The 6 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.