Minimum deceleration to prevent a collision

[haruspex]

By the way, we see that if $d$ is lesser than a certain value the back car will hit the front one. We still don't know if the front car is in motion on not when the back car hits it. Can you figure out the condition that the back car collides the front one, when front car is still moving?

That goes back to post #1.

 

[PKM]

That goes back to post #1.

Why?
I actually meant to find the condition that the collision occurs before car 1 comes to rest. That wasn't asked in the question. The collision may occur when car 1 has already stopped, or still running. I asked about finding out a constraint to ensure which one occurs. Does it make sense now?

 

[haruspex]

Why?

Because the approach there assumed the two SUVAT equations quoted applied up to the point of collision. That was not valid for the original question but is true for the question you pose.

 

[Jenny Physics]

By the way, we see that if $d$ is lesser than a certain value the back car will hit the front one. We still don't know if the front car is in motion on not when the back car hits it. Can you figure out the condition that the back car collides the front one, when front car is still moving?

The time it takes for the front car to stop is $t_{1}=\frac{v_{1}}{2d}$. Assuming that both cars are decelerating the collision time is $t_{collision}=\frac{L}{v_{2}-v_{1}}$. This time is smaller or equal to $t_{1}$ if $d< \frac{v_{1}}{v_{1}+v_{2}}d_{critical}$ where $d_{critical}=\frac{v_{2}^{2}-v_{1}^{2}}{2L}$.
This is the range of $d$ for which the back car will hit the front car while the front car is still moving. This means that for $\frac{v_{1}}{v_{1}+v_{2}}d_{critical}\leq d<d_{critical}$ the back car will collide with a stopped front car.