Question

Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

Short answer

Dot product of the velocity and acceleration vectors is 0; Hence they are orthogonal.

Step-by-step solution

Orthogonal vectors

If two vectors are orthogonal that means they are perpendicular to each other. Also, if the dot product is zero i.e.,\({\rm{a(t)}}{\rm{.v(t) = 0}}\)the two vectors are said to be orthogonal vectors.

Finding dot product of velocity and acceleration vector

Let \(r(t)\)be the position vector.

As particle is moving with constant speed means-

\(\begin{aligned}{l}|r'(t)| = c\\|r'(t){|^2} = {c^2} = k\end{aligned}\)

\(\begin{aligned}{c}\frac{d}{{dt}}r'(t) \cdot r'(t) = \frac{d}{{dt}}k\\r'(t) \cdot r''(t) + r'(t) \cdot r''(t) = 0\\2r''(t) \cdot r'(t) = 0\\r''(t) \cdot r'(t) = 0\end{aligned}\)

As Velocity vector is the derivative of the position vector and acceleration vector is the second derivative of the position vector or derivative of the velocity vector.

i.e., \(r''(t) = a(t)\)and \(r'(t) = v(t)\).

We get,

\(a(t) \cdot v(t) = 0\)

Since if dot product is zero, then those two vectors are orthogonal.

Therefore, the velocity and acceleration vectors are orthogonal.