This is Ant on a rubber rope problem

ANSWER IN SHORT-
The worm will eventually reach the end.
The key to understand this is that the proportion between the total length of the  rubber Band and the length walked by the worn can only increase with time, as it will eventually approach one.

EXPLAINATION-
The problem seems impossible because we are looking worm and the rubber Band moving independently.

Things become a little more imaginable – “When we realize that the Worm is on the rubber Band, and the portion of the rubber Band behind the worm also stretches just as the portion in front of the worm does,”

The mathematics are complex, but think of the entire picture of the worm and the rubber Band.

At 0 Hour, the worm is at the very end of the rubber Band, and has 100 percent of the rubber Band in front of it.
At 1 Hour, it’s true that the Worms’s task is considerably increased, but it doesn’t have one hundred percent of the rubber Band in front of it.
And that tiny percentage of the rubber Band that the anworm has traversed will stretch in proportion, just like the rest of the rubber Band.

So, Percentage of distance to be covered is increasing in same proportion to the Distance already covered.
i.e.

Percent of Distance traveled is increasing     DIRECTLY PROPORTIONAL to        Increase in distance to be traveled.

MATHEMATICAL APPROACH-

Yes, the ant will reach the car. Calculus tells us exactly when:

if a < 0, then exp(c/a) < 1 and t < 0, which means no solution. The faster the car is, the greater [exp(c/a)]/c is and thus, the longer it is for the ant to reach the car.