Tangential and normal components of velocity

Tangential and normal components

When a particle moves along a curved path, its velocity vector is always tangent to the path. Choose a moving frame at the particle with unit vectors e_t (along the tangent, in the direction of motion) and e_n (perpendicular to it, toward the centre of curvature).

• Tangential component of velocity: v_t = v = ds/dt, the speed of the particle along the path.
• Normal component of velocity: v_n = 0, because the velocity has no component perpendicular to the path.

So v = v e_t.

For the acceleration the normal and tangential components are both present:

• a_t = dv/dt (rate of change of speed)
• a_n = v²/ρ (rate of change of direction, ρ = radius of curvature)

Examples

1. Car on a circular track of radius 100 m moving at constant 20 m/s: v_t = 20 m/s, v_n = 0, a_t = 0, a_n = 20²/100 = 4 m/s² toward the centre.
2. Car on the same track speeding up at 2 m/s² when its speed is 20 m/s: a_t = 2 m/s², a_n = 4 m/s², total a = √(2² + 4²) = 4.47 m/s².
3. A projectile at its highest point: velocity is horizontal (tangent), v_t = v_x; the whole acceleration g is normal, so ρ = v_x²/g.