two rocks of unequal mass have the same kinetic energy and are sliding across a flat, level ice surface in the same direction. If the each is subjected to the same retarding force, how will their stopping distances compare?

To compare the stopping distances of two rocks of unequal mass that have the same kinetic energy and are subjected to the same retarding force while sliding on a flat, level ice surface, we need to understand the relationship between kinetic energy, mass, force, and stopping distance.

1. Kinetic Energy (KE): The kinetic energy of an object is given by the equation KE = (1/2) * mass * velocity^2. It depends on mass and velocity, where velocity is the speed at which the object is moving.

2. Force (F): The stopping force acting on an object determines its deceleration, which is given by Newton's second law of motion as F = mass * acceleration. In this case, the force acting on each rock is the same.

3. Stopping Distance: The distance traveled by an object during deceleration until it comes to rest is called the stopping distance. It depends on the initial velocity, deceleration, and time taken to stop.

Since the rocks have the same kinetic energy, their velocities may differ based on their masses. However, let's assume their initial velocities are the same for simplicity. Now, since both rocks experience the same retarding force, we can conclude that their decelerations will also be the same.

The stopping distance is directly proportional to the square of the initial velocity (v₀) and inversely proportional to the acceleration (a). Mathematically, stopping distance (d) is given by the equation d = (v₀^2) / (2a).

Since the initial velocity is the same and the acceleration is the same for both rocks, the stopping distance will be the same for both rocks.

In summary, when two rocks of unequal mass have the same kinetic energy, slide on a flat, level ice surface, and experience the same retarding force, their stopping distances will be the same.