A particle is st in motion at the moment t=0, at the velocity v0 wher upon its velocity begins changing with time in accordance with the...

1) We know that the space is the product of velocity and
time.

r = v*t

We'll
differentiate both sides:

dr =
vdt

To calulate r = Integral
vdt

We'll substitute v by the given
expression:

Integral vdt = Int
v0(1-t/b)dt

Int v0(1-t/b)dt =
v0*Int(1-t/b)dt

v0*Int(1-t/b)dt = v0*(Int dt -Int
tdt/b)

v0*(Int dt -Int tdt/b) = v0*t - (v0/b)*t^2/2 +
C

r = v0*t - (v0/b)*t^2/2 +
C

2) We'll calculate the distance covered by
the particle in time:

s = Int
vdt

In this case v =
v0*|1-t/b|

If t<b, we'll have v = v0*(1 -
t/b)

If t>b, we'll have v = v0*(t/b -
1)

If
t<b

s = Int v0*(1 -
t/b) dt

s = v0*t - (v0/b)*t^2/2 +
C

If
t>b

s = Int v0*(t/b - 1)
dt

s = v0*b{1+[1-(t/b)]^2}/2 +
C