The function a(t)=t^(1/2)−t^(−1/2) m/s^2 represents the acceleration of a particle moving along a horizontal axis. At time t=0, its velocity is equal to (4/3)m/s, and its position is equal to (-4/15) m.
Part A: Find the velocity function v(t) of the particle.

Part B: Find the position function s(t) of the particle.

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

v(t) = ⅔(t^1.5) - 2(t^0.5) + 4/3

s(t) = (4/15)(t^2.5) - (4/3)(t^1.5) + (4/3)t - 4/15

Step-by-step explanation

a(t) = t^½ - t^-½

Integrate for v(t)

(+1 to the power, divide by the new power)

v(t) = ⅔(t^3/2) - 2(t^½)n+ c

At t = 0, v = 4/3

4/3 = 0 + 0 + c

v(t) = ⅔(t^1.5) - 2(t^0.5) + 4/3

Integrate for s(t)

s(t) = (⅔)(⅖)(t^5/2) - 2(2/3)(t^3/2) + (4/3)t + c

s(t) = (4/15)(t^2.5) - (4/3)(t^1.5) + (4/3)t + c

At t = 0, s = -4/15

-4/15 = 0 - 0 + 0 + c

c = -4/15

s(t) = (4/15)(t^2.5) - (4/3)(t^1.5) + (4/3)t - 4/15

last one.

step-by-step explanation:

last one.

generally speaking, the others follow some sort of pattern that correlates to either linear (looks like a straight line) or sinusoidal (looks like an s).

the last one is just a random cloud of points that doesn't correlate to any particular known order.

1)is not similar since corresponding sides are not in the same ratio

2)are similar since corresponding sides are in the same ratio and corresponding angles are equal

hope it will

#faazmessi