Car A is traveling a distance d behind Car B. Initically both cars are traveling at the same speed of 60 ft/s. Suddenly Car B applies the brakes, causing Car B to decelerate at 8ft/s2. It takes the driver of Car A 0.75 seconds to react, and when she applies her brakes Car A decelerates at 20ft/s2.
What is the initial minimum distance d between the cars so as to avoid a collision?
The Attempt at a Solution
FIrst thing I did was solve for xf.b=0+60(0.75)+(-8*.5*0.752) = 42.75ft
Secondly, I solved for xf.a=0+60(0.75) = 45ft
So I know after 0.75 seconds, the distance d = c+2.25. Now I need to solve for c.
I know that A will continually get closer to B until their velocities are equal again so I then set vf=vo+a*t equal to each other for each car (once I found final velocity of B after 0.75 seconds) and solved for t, which was 0.5 seconds.
After that I solved xf=xi+vo*t+(1/2)a*t2 for each car again, getting Δxa=27.5, Δxb=26. 27.5-26 = c, which i then substituted in d = c + 2.25 to get the answer.
I got the right answer, but I am really unsure about the process. I thought my way through it but it was really roundabout. Is there a more logical process to solving this?