An object moves along a coordinate line with acceleration a(t)=(t+1)^-1/2 units/second.
a. Find velocity function given that initial velocity is 2 units/second.
b. Find the position that the initial velocity is 1 unit/second and the initial position of the object is 0.
For part A I understand that a(t)=v'(t) and I need to differentiate. Therefore I found that v(t)=2(t-1)^1/2 + c
However, I don't know if this is right, where does the vo+at come in? And what about the initial velocity of 2 units/second?
For both parts would appreciate steps as to how to go about it, not just looking for the answer. Looking for feedback. Thank you in advance.
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acceleration means the differentiation of velocity.
`(dv(t))/dt = a(t)`
To get the velocity function we need to integrate the function.
`dv(t) = a(t)*dt`
`intdv(t) = int a(t)dt`
`v(t) = int a(t)dt + C` where C is a constant.
`= int a(t)dt + C`
`= int (t+1)^(-1/2)dt + C`
`= (t+1)^(-1/2+1)/(-1/2+1) + C`
`= 2(t+1)^(1/2) + C`
It is given that initially the velocity is 2. That means at time t = 0 the velocity is 2 or v(0) = 2
`v(t) = 2(t+1)^(1/2) + C`
`v(0) = 2(0+1)^(1/2) + C`
`2 = 2+C`
`C = 0`
Velocity function `v(t) = 2(t+1)^(1/2)`
you can use `(ds(t))/dt = v(t)` combination to get the position function of the object. Use the same procedure as the velocity function.
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