**Car Stopping Distance and Kinetic Energy**

To understand how doubling the kinetic energy of a car affects its stopping distance, we need to consider the relationship between kinetic energy, friction, and stopping distance.

**1. Initial Scenario**

Let's assume a car of mass m moving at a speed v. When the brakes are applied, the friction between the tires and the road provides a force that opposes the car's motion. This frictional force acts over a distance x to bring the car to a stop.

**2. Kinetic Energy and Stopping Distance**

The kinetic energy of an object is given by the equation:

KE = (1/2)mv^2

This equation tells us that the kinetic energy of the car is directly proportional to its mass and the square of its velocity. When the car is brought to a stop, all of its initial kinetic energy is converted into other forms, such as heat and sound.

The stopping distance, x, is determined by the work done by the frictional force to bring the car to a stop. The work done is given by the equation:

Work = Force * Distance

In this case, the force is the frictional force opposing the car's motion, and the distance is the stopping distance.

**3. Doubling the Kinetic Energy**

If the kinetic energy of the car is doubled, the new kinetic energy, KE', can be expressed as:

KE' = 2KE

Since KE = (1/2)mv^2, doubling the kinetic energy yields:

KE' = (1/2)m(2v)^2 = 2mv^2

Therefore, the new kinetic energy is four times the initial kinetic energy.

**4. Effect on Stopping Distance**

To determine the effect on the stopping distance, we need to consider the work done by the frictional force. Since work is equal to force times distance, and the force remains constant, we can express the work done in terms of the stopping distance:

Work = Force * Distance = (Frictional Force)(Stopping Distance)

Since the work done is equal to the initial kinetic energy (KE), and the new kinetic energy (KE') is four times the initial kinetic energy, the work done in the new scenario can be expressed as:

Work' = 4KE = (Frictional Force)(New Stopping Distance)

Comparing the two work equations, we can deduce the relationship between the initial stopping distance (x) and the new stopping distance (x'):

(Frictional Force)(Stopping Distance) = (Frictional Force)(New Stopping Distance)

This equation implies that the stopping distance has to quadruple when the kinetic energy is doubled:

Stopping Distance' = 4(Stopping Distance) = 4x

Therefore, doubling the kinetic energy of the car will result in a stopping distance that is four times greater than the initial stopping distance.

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