Question

Find the position vector of a particle which satisfies a(t) = with v(0) = <1, 1> and r(0) = <2, -1>. ???

Answer 1

a(t) = what?

Answer 2

sorry , a(t) <sin t, e^t>

Answer 3

I integrated twice a(t) to get position but when I set t = 0 it does not give me what it says there

Answer 4

since a(t) = v'(t),
\[v'(t) = <\sin(t),e^t>\]
\[v(t) = <-\cos(t) + c_1, e^t + c_2>\]
using the initial conditions,
\[v(0) = <-1+c_1,1+c_2> = <1,1>\]
so
\[c_1=2,c_2=0\]
and
\[v(t) = <2-\cos(t),e^t>\]
integrating again,
\[r(t) = <2t - \sin(t) + c_3,e^t+c_4>\]
\[r(0)=<c_3,1+c_4> = <2,-1>\]
so
\[c_3=2,c_4=-2\]
and finally
\[r(t)=<2t-\sin(t)+2,e^t-2>\]

Answer 5

perfect! Thank you so much!