I have a problem that I am working on called "Calvin and Phoebe's Velocity Problem."
The two characters distance equations are given as:
Phoebe: s(t)=e^(t) - 1
s is feet and t is seconds.
Part one was to decided which one had the greatest initial acceleration. I figured it was Phoebe by entering 1 into each equation
Next was to find when Calvin catches up with phoebe. This was there I had problems. I was not able to do this algebraically, but finding the intersection graphically I got t=1.545. But I need to know how to do it algebraically.
I also have to find the maximum distance between the two cars on the interval (1,1.545). I started by setting D=(e^(t) - 1) - (t^3)
I thought I had to then take the derivatives and set that equal to zero, then solve for t, but when I did that i get stuck when i reach t=ln(3t^2). Where would I need to go from there? Solving graphically I got .91 ft
I also have to find maximum difference in velocity, which I am guessing is done similarly to the one previously. Again I get stuck when setting the derivatives equal to 0. Graphically I got .2157 ft/sec.
It would be greatly appreciated if anybody could help me with what I am doing wrong, how to find these answers algebraically. Thank you.