Final answer:
To find the position vector of a particle with a given acceleration and initial conditions, integrate the acceleration to find velocity and integrate velocity to find position. The specific position vector in this case is r(t)=(8/3)t^3i - (sin(t)-1)j - (1/4)cos(2t)k.
Step-by-step explanation:
To find the position vector of a particle that has the given acceleration and the specified initial velocity and position, we can integrate the acceleration function, a(t), with respect to time to find the velocity function, and then integrate again to find the position function, r(t). The given acceleration vector is a(t)=16ti+sin(t)j+cos(2t)k, with initial velocity v(0)=i and initial position r(0)=j. First, we find the velocity function by integrating the acceleration function with respect to time:
- Integrate 16t with respect to t to get 8t2i
- Integrate sin(t) with respect to t to get -cos(t)j
- Integrate cos(2t) with respect to t to get (1/2)sin(2t)k
Now we add the constant of integration, which is the initial velocity vector v(0)=i:
V(t)=8t2i - cos(t)j + (1/2)sin(2t)k + i
To find r(t), we integrate the velocity function V(t):
- Integrate 8t2i with respect to t to get (8/3)t3i
- Integrate -cos(t)j with respect to t to get -sin(t)j
- Integrate (1/2)sin(2t)k with respect to t to get -(1/4)cos(2t)k
Adding the constant of integration, which is the initial position vector r(0)=j:
r(t)=(8/3)t3i - sin(t)j - (1/4)cos(2t)k + j
The position vector of the particle is given by:
r(t)=(8/3)t3i - (sin(t)-1)j - (1/4)cos(2t)k