Deceleration to equal position and velocity

Question

Hi guys!
I have a interesting math problem and I am by no means a math expert!

So if i have one vehicle traveling at 120 km/h or 33.333 m/s, and another vehicle traveling at 84 km/h or 23.333 m/s and is 60 m in front of the other vehicle.

How would I calculate the minimum linear deceleration rate to ensure that the two vehicle's position would be the same, at the same velocity?

I literally had to guess and add values to come close to such a rate:

33.333(11,862396)+1/2(-0.843)(11,862396)^2 for the first vehicle.

= 336,101 m

23.333(11,862396) + 60 for the second vehicle

= 336,789 m

I am just using these equation as a example, as you can see, the first vehicle will -with that deceleration-, never reach the second vehicle (technically).

I filled in the time and deceleration, but that is where i need help. I don't know how to get the minimum value that the deceleration can be to make sure that the two vehicles are in the same place, with the same speed.

Answer 1

Equate the distances traveled:

 

33t + 1/2at² = 23t + 60

 

We also have a = (23 - 33)/t = -10/t ,  so we can substitute this into the first equation to eliminate a:

 

33t + 1/2(-10/t)t² = 33t - 5t = 23t + 60

 

Then solve for t = 12, then a = -10/12 = -8.3 m/s²

 

Repeat without rounding if you need a more precise answer.