Describe the motion of a particle when the tangential component of acceleration is 0.

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Mary Moen · Accepted Answer

Let r(t) be a vector-valued function that denotes the position of an object as a function of time. Then $a (t) = r (t)$ is the acceleration vector. The tangential component of acceleration $a_{T}$ is given by the formula
${\vec{a}}_{r} = \vec{a} \cdot \vec{T} = \frac{\vec{v} \cdot \vec{a}}{∥ \vec{v} ∥}$
Where T(t) is the unit tangent vector to the curve defined by r(t).
Now given in the question the tangential component of acceleration is 0, then
${\vec{a}}_{r} = \vec{a} \cdot \vec{T} = \frac{\vec{v} \cdot \vec{a}}{∥ \vec{v} ∥} = 0$
$\frac{\vec{v} \cdot \vec{a}}{∥ \vec{v} ∥} = 0$
$\vec{v} \cdot \vec{a} = 0$
$∣ v ∥ a ∣ \cos θ = 0$
$\cos θ = \cos 90$
$θ = 90$
Hence the velocity (Motion vector) is perpendicular to the acceleration of the given position vector.

Describe the motion of a particle when the tangential component

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