A person on a hang glider is spiraling upward due to rapidly rising air. The acceleration of the glider is given by a vector function a(t)= -3cos(t) i - 3sin(t) j +2 k. The glider departed (t=0) from the point (3,0,0) with a velocity of vector function v(t)= 3 j. Find the glider's position as a function of time, t.
To find the position as a function of time, we need to integrate the acceleration vector twice.
First, we integrate the acceleration vector to get the velocity vector:
v(t) = ∫ a(t) dt = ∫ (-3cos(t) i - 3sin(t) j +2 k) dt = -3sin(t) i + 3cos(t) j + 2t k + C1
where C1 is a constant of integration. We use the initial velocity v(0) = 3 j to find C1:
v(0) = 3 j = 0 i + 3 j + 0 k + C1
C1 = 3 j
So, v(t) = -3sin(t) i + 3cos(t) j + 2t k + 3 j
Next, we integrate the velocity vector to get the position vector:
r(t) = ∫ v(t) dt = ∫ (-3sin(t) i + 3cos(t) j + 2t k + 3 j) dt
= 3cos(t) i + 3sin(t) j + t^2 k + 3t j + C2
where C2 is another constant of integration. We use the initial position r(0) = (3,0,0) to find C2:
r(0) = (3,0,0) = 3 i + 0 j + 0 k + C2
C2 = (0,0,0)
So, the position vector as a function of time is:
r(t) = 3cos(t) i + 3sin(t) j + t^2 k + 3t j
Therefore, the glider's position as a function of time is given by the vector-valued function r(t) = <3cos(t), 3sin(t) + 3t, t^2>.