Projectile given in i/j vector form - find the time or height at which the velocity is perpendicular to the initial launch velocity
IAL M2 · Vectors / projectiles · appears in 3 of 25 sittings, 2023-2024
A particle is projected with a given initial velocity in i/j vector form and moves freely under gravity. Find the time (or, from that, the height above the ground) at the instant when the particle's velocity is perpendicular to its initial velocity.
A projectile whose velocity must be perpendicular to its own velocity at another instant appears in 5 of 25 M2 sittings. In 2 of them (2020-October, 2021-June) the other instant is a later point or the landing, not the launch, so the same scalar-product step is applied to a velocity the student first has to find.
The method
4 marks at its longest. Each step below is a mark the examiner is looking for.
Form the perpendicularity condition: the dot product of the initial velocity vector and the velocity vector at time t is zero; condone sign errors in setting this up (1 mark)
Solve the resulting equation for the vertical velocity component, or directly for t, depending on what the question asks for (1 mark)
Where a height is asked for: use suvat or an energy equation, dependent on the previous method mark, to convert the found velocity/time into a height above the ground (2 marks)
What changes between years
Everything below varies from sitting to sitting. The method above does not.
whether the question asks for the time directly (2023-October) or requires a further step to convert that into a height above the ground (2023-June, 2024-January)
the launch height (a platform above ground level in 2023-June vs ground level in the other two)
the initial velocity components and the resulting numeric answer
whether a second particle is involved in the surrounding question (2023-October frames this as one particle, Q, within a multi-particle question; the other two use a single particle P)
Where the marks go
Instructions the examiners are given, repeated across sittings.
Sign errors in setting up the dot-product condition are explicitly condoned at the method-mark stage (3 sittings)
The height-conversion step is a dependent method mark - it is only awarded once the perpendicularity condition has already been solved correctly (2 sittings)
Using the velocity component with the wrong sign is explicitly flagged as not equivalent working, so is not simply a follow-through error (1 sittings)
Where it appeared
2023-June, 2023-October, 2024-January.
GOAL-15 Phase 3 recounted this over all 25 M2 sittings (it had been counted over 11). No sitting outside the three already on record asks for perpendicularity to the initial velocity.