# A particle moves with acceleration `a(t)` m/s^2 along an s-axis and has velocity `v_0` m/s at time `t=0` . Find the displacement AND the distance traveled by the particle during the given...

A particle moves with acceleration `a(t)` m/s^2 along an s-axis and has velocity `v_0` m/s at time `t=0` . Find the displacement AND the distance traveled by the particle during the given time interval.

`a(t)=9 ; `

`v_0=-1 ; `

`0lt=tlt=2`

*print*Print*list*Cite

`a(t) = (dV(t))/(dt)`

`dV(t) = a(t)dt`

`intdV(t) = inta(t)dt`

`V(t) = int9dt`

`V(t) = 9t+C`

`V(0) = 9xx0+C`

`-1 = C`

`V(t) = 9t-1`

`V(t) = (dS(t))/(dt)`

`S(t) = intV(t)dt`

`S(t) = int(9t-1)dt`

`S(t) = 9t^2/2-t+D` where D is a constant.

It is given that the time interval is `0<=t<=2`

Displacement of particle `= S(2)-S(0)`

Displacement of particle `= (9xx4/2-2) = 16m`

For distance travelled;

`S(t) = int_0^2(9t-1)dt`

`S(t) = [9t^2/2-t]_0^2`

`S(t) = (9xx4/2-2) = 16`

*So the displacement and distance travelled by the particle is 16m.*

The particle moves along a straight line with initial velocity `v_0` = -1 m/s and acceleration a(t) = 9 m/s^2. The distance traveled by the particle during a duration of time equal to t is `s = v_0*t + (1/2)*a(t)*t^2` .

Substituting the given values the displacement of the particle in the period from t = 0 to t = 2 is:

(2 - 0)*9 + (1/2)*9*(2 - 0)

=> 18 + 9

=> 27

**The particle travels 27 m in the given interval.**