1D Kinematics: Distance between 2 cars

[Nikstykal]

Homework Statement

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?

Homework Equations

x_f=x_i+v_o*t+(1/2)a*t²
v_f=v_o+a*t

The Attempt at a Solution

FIrst thing I did was solve for x_f.b=0+60(0.75)+(-8*.5*0.75²) = 42.75ft
Secondly, I solved for x_f.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 v_f=v_o+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 x_f=x_i+v_o*t+(1/2)a*t² for each car again, getting Δx_a=27.5, Δx_b=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?

 

[drvrm]

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?

pl. explain your doubts about your calculation .

 

[Orodruin]

Which part are you unsure about? Your reasoning seems solid.

 

[Nikstykal]

Thank you for the quick reply. I feel like I was just solving things randomly, maybe it is just because this stuff is new and I don;t know exactly what I am doing yet. I wasn't confident in my approach until I got the right answer.