Answer:
see below
Step-by-step explanation:
a(t)=t^(1/2)−t^(−1/2)
We integrate to find the velocity
v(t) = integral t^(1/2)−t^(−1/2) dt
= t ^ (1/2 +1) t ^ (-1/2 +1)
------------ - ----------------- + c where c is the constant of integration
3/2 1/2
v(t) = 2/3 t^ 3/2 - 2 t^ 1/2 +c
We find c by letting t=0 since we know the velocity is 4/3 when t=0
v(0) = 2/3 0^ 3/2 - 2 0^ 1/2 +c = 4/3
0+c =4/3
c = 4/3
v(t) = 2/3 t^ 3/2 - 2 t^ 1/2 +4/3
To find the position function we need to integrate the velocity
p(t) = integral 2/3 t^ 3/2 - 2 t^ 1/2 +4/3 dt
2/3 t ^ (3/2 +1) 2 t ^ (1/2 +1) 4/3t
------------ - ----------------- + ------------- + c
5/2 3/2 1
p(t) = 4/15 t^ 5/2 - 4/3t ^ 3/2 + 4/3t +c
We find c by letting t=0 since we know the position is -4/15 when t=0
p(0) = 4/15 0^ 5/2 - 4/3 0 ^ 3/2 + 4/3*0 +c = -4/15
0 +c = -4/15
c = -4/15
p(t) = 4/15 t^ 5/2 - 4/3t ^ 3/2 + 4/3t -4/15
