Question

Assume that a particle is at the origin when the time t is at the origin

Determine the position vector of the velocity of the particle at time t given the velocity as

V(t)=e^-ti+5e^-2tj

Answer 1

$$r(t)=\int V(t)dt=\int (e^{-t}i+5e^{-2t}j)dt$$

$$=-e^{-t}i-\dfrac{5}{2}e^{-2t}j+r_0$$

$$r(0)=-i-\dfrac{5}{2}j+r_0=\pmb{0}$$

$$r(0)=i+\dfrac{5}{2}j+r_0$$

$$r(t)=(1-e^{-t})i+\dfrac{5}{2}(1-e^{-2t})j$$