The velocity as a function of time for an asteroid in the asteroid belt is given by
v(t)= v0e^(t/t0) I + v0t/2t0 j

with vo and to constants.

Use ti = 0 as the initial time, and tf = 1.5 to as the final time.

The values for the constants that you will use are:

vo = 4.5 m/s
to = 80 s

Find the displacement of the asteroid.

I don't understand why you introduced the factors t_i and t_f, but be that as it may...

Given the velocity functions you can find the displacements in the i and j directions by integrating:


x = \displaystyle \int_{t_0}^{t} v_i(t) dt = \displaystyle \int_{t_0}^{t} e^{(t/t_0)} dt\\


y = \displaystyle \int_{t_0}^{t} v_j(t) dt = \displaystyle \int_{t_0}^{t} (\frac {v_o}{2 t_o}) t dt\


Hope this helps.