Final answer:
The position vector of a particle can be found by integrating the given acceleration function twice with respect to time. First, we integrate the acceleration function to find the velocity function, and then we integrate the velocity function to find the position function. Using the given initial velocity and position, we can solve for the constants and determine the position vector of the particle.
Step-by-step explanation:
The position vector of a particle can be found by integrating the given acceleration function twice with respect to time. First, we integrate the acceleration function to find the velocity function:
v(t) = ∫(a(t) dt) = ∫(19t i + et j + e^(-t) k dt) = 9.5t^2 i + et j - e^(-t) k + C1
Then, we integrate the velocity function to find the position function:
r(t) = ∫(v(t) dt) = ∫((9.5t^2 i + et j - e^(-t) k) dt) = 3.17t^3 i + et^2/2 j + e^(-t) k + C1t + C2
Using the given initial velocity and position, we can solve for the constants C1 and C2:
v(0) = k => 9.5(0)^2 i + e(0) j - e^(-0) k + C1 = k => C1 = 0
r(0) = j + k => 3.17(0)^3 i + e(0)^2/2 j + e^(-0) k + 0 + C2 = j + k => C2 = 1
Therefore, the position vector of the particle is:
r(t) = 3.17t^3 i + et^2/2 j + e^(-t) k + t + 1