a particle is moving with the given data. find the position of the particle, ( s(t) ).\n( v(t)=9 cos (t)+7…

a particle is moving with the given data. find the position of the particle, ( s(t) ).\n( v(t)=9 cos (t)+7 sin (t), quad s(0)=8 )\n( s(t)= )

a particle is moving with the given data. find the position of the particle, ( s(t) ).\n( v(t)=9 cos (t)+7 sin (t), quad s(0)=8 )\n( s(t)= )

Answer

Explanation:

Step1: Integrate the velocity function

The position function (s(t)) is the integral of the velocity function (v(t)). [ \begin{align*} s(t)&=\int v(t)dt\ &=\int(9\cos(t)+7\sin(t))dt\ &=9\int\cos(t)dt + 7\int\sin(t)dt\ &=9\sin(t)-7\cos(t)+C \end{align*} ]

Step2: Use the initial condition to find (C)

Given (s(0) = 8), substitute (t = 0) into (s(t)): [ \begin{align*} s(0)&=9\sin(0)-7\cos(0)+C\ 8&=9\times0 - 7\times1+C\ 8&=- 7+C\ C&=15 \end{align*} ]

Answer:

(s(t)=9\sin(t)-7\cos(t)+15)