Apply the Motion Equations Component by Component
When position, velocity or acceleration are given as vectors, every kinematic equation still holds along each axis separately, and magnitudes are combined only at the end.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Constant vector acceleration
A particle with initial velocity (3, 4) and acceleration (0.4, 0.3) has velocity (3 + 4, 4 + 3) = (7, 7) after 10 s, so its speed is 7 sqrt(2). A body starting from rest with accelerations 6 m/s² along x and 8 m/s² along y has a resultant acceleration of 10 m/s² along a fixed direction, so after 4 s its distance from the origin is (1/2)(10)(16) = 80 m.
Position vectors that rotate
Differentiating gives a velocity of constant magnitude ω perpendicular to r, and a second derivative of -ω² r, so the acceleration points toward the centre (centripetal).
When the velocity and acceleration vectors are perpendicular at every instant, the speed is constant while the direction changes.
OlympIQ exam lens
- ✓Apply v = u + at and s = ut + (1/2)at² to each component separately
- ✓Speed = magnitude of the velocity vector; distance from the origin = magnitude of the position vector
- ✓Constant |r| means circular motion; the acceleration points to the centre, a = -ω² r
- ✓Speed = ω x radius for uniform circular motion
Do not add the magnitudes of the components. Speed is the square root of the sum of the squares of the velocity components, and a body with equal components along x and y has speed sqrt(2) times one component.
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