Velocity and acceleration of particle on curve

The position vector is r = a cos t i + a sin t j + bt k (a circular helix). Velocity v = dr/dt and acceleration f = d²r/dt².

Velocity

v = dr/dt = −a sin t i + a cos t j + b k

Acceleration

f = d²r/dt² = −a cos t i − a sin t j

At t = 0

v = −a(0) i + a(1) j + b k = a j + b k
f = −a i

Speed = |v| = √(a² + b²).

At t = π/2

v = −a(1) i + a(0) j + b k = −a i + b k
f = −a(0) i − a(1) j = −a j

Speed = |v| = √(a² + b²).

Answer:

• t = 0: velocity = a j + b k, acceleration = −a i.
• t = π/2: velocity = −a i + b k, acceleration = −a j.

The speed √(a² + b²) is constant, and the acceleration has constant magnitude a, directed towards the z-axis.