an object starts from rest and accelerates uniformly at a rate of 2m/s^2 for 20s after that it maintains a constant speed for 200s then it starts to retard and comes to rest in 100s find the maximum velocity reached, retardation ,the total distance travelled and the average velocity of the object

First case : the body is accelerating form rest

here

u = 0

a₁ = 2 m/s²

t₁ = 20m

now, final velocity

v₁ = u + a₁t₁

so,

v₁ = 0 + 2x20

thus,

v₁ = 40 m/s

and distance travelled

s₁ = ut₁ + (1/2)a₁t₁²

or

s₁ = 0 + (1/2)x2x20²

thus,

s₁ = 400m

.

Second case : the body is moving with no acceleration

here

v₂ = v₁ = 40 m/s

a₂ = 0

t₂ = 200s

now,

distance travelled

s₂ = v₂.t₂

or

s₂ = 40 x 200

thus,

s₂ = 8000m

.

Third case : the body is retarding to rest

u₃ = v₂ = 40 m/s

v₃ = 0

t₃ = 100s

so, retardation

a₃ = (v₃ - u₃) / t

or

a₃ = (0 - 40) / 100

thus,

a₃ = -0.4 m/s

now, distance travelled

s₃ = u₃t₃ +(1/2)a₃t₃²

or

s₃ = 40x100 + (1/2)x-0.4x100²

or

s₃ = 4000 - 0.2x10000

thus,

s₃ = 2000m

.

now,

max velocity = 40 m/s

retardation = -0.4 m/s²

total distance travelled = 400m + 8000m + 2000m = 10400m

average velocity = net displacement / total time = S/T =  10400 / (20+200+100)

= 10400 / 320

or

average velocity = 32.5 m/s