Two lovers are parked 10.0m from the edge of a cliff in a car whose mass, including that of the occupants is 100kg. a jealous suitor ties a rope to the car's bumper and a 50. kg rock to the other end of the rope. he then lower the rock over the edge of the cliff, and the car, which is in neutral, accelerates toward the edge.

b) what is the acceleration of the car towards the edge?
c) how long do the lovers have to apply the brakes before they go over the edge?

I will solve this assuming: the rock is lowered at constant speed; the surface is level, and there is no friction 

then, the tension in the rope equals the weight of the rock, so T= 50kg x 9.8m/s/s 

T=490N; this is the tension in the rope and the force that pulls on the car, so we have 

F = ma => a=F/m=490N/1000kg = 0.49m/s/s 

if you are initially at rest and accelerate at 0.49m/s/s, it will take a time given by 

dist = 1/2 at^2 => t=sqrt[2d/a] to travel 10 m: 

t=sqrt[2x10m/0.49m/s/s] = 6.4s which is the time they have to react.

Hello there.
c) How long do the lovers have to apply the brakes before they go over the edge?

 6.4s