As shown in the figure, a particle is moving with constant speed $$\pi$$ m s$$^{-1}$$. Considering its motion from A to B, the magnitude of the average velocity is

given
speed = π m/s (constant)
motion is along a circle
angle from A to B = 120°

we need average velocity = displacement / time

step 1: understand what displacement means

even though motion is along a curve, displacement is always the straight line from A to B

so here displacement = chord AB

step 2: find displacement (chord AB)

draw the triangle formed by center O, A, and B

OA = OB = R
angle AOB = 120°

chord formula:

AB = 2R sin(θ/2)

θ = 120° → θ/2 = 60°

AB = 2R sin60°
= 2R × (√3/2)
= R√3

so displacement = R√3

step 3: find time taken

the particle moves along the arc from A to B

arc length = Rθ

convert 120° into radians:

120° = 2π/3

so arc length = R × (2π/3)

speed = π

time = distance / speed

t = [R(2π/3)] / π

cancel π:

t = 2R/3

step 4: compute average velocity

average velocity = displacement / time

= (R√3) / (2R/3)

now simplify:

= R√3 × (3 / 2R)

R cancels:

= (3√3)/2

= 1.5√3 m/s

final answer: 1.5√3 m/s