Calculating the Average Frictional Force Stopping a Car

When a high-mass car, with a mass of 2000 kg and an initial speed of 20 meters per second, comes to a rest over a distance of 100 meters, what is the average frictional force that brought it to a halt? This article will guide you through the process of calculating this force using the principles of physics and applied mathematics. Let's delve into the detailed steps to find the solution.

Step 1: Calculate the Initial Kinetic Energy of the Car

The kinetic energy (KE) of an object is given by the formula:

KE frac{1}{2}mv^2

where:

m is the mass of the car (2000 kg), v is the initial velocity of the car (20 m/s).

Substituting the given values:

KE frac{1}{2} times 2000 text{kg} times (20 text{m/s})^2

KE 1000 times 400 400000 text{J}

Step 2: Calculate the Work Done by the Frictional Force

The work done by the frictional force W is equal to the force F multiplied by the distance d over which it acts, according to the work-energy principle:

W F cdot d

In this case, the work done by the frictional force is equivalent to the negative change in kinetic energy, as it is acting in the opposite direction of motion:

W -KE

Thus, we have:

-400000 text{J} F cdot 100 text{m}

Step 3: Solve for the Frictional Force F

Rearranging the equation to find F:

F frac{-400000 text{J}}{100 text{m}} -4000 text{N}

The negative sign indicates that the force is acting in the opposite direction to the motion of the car. Therefore, the average frictional force tending to stop the car is 4000 N.

Using the Alternative Method

Alternatively, you can use the equation of motion v^2 u^2 2as to find the deceleration. Here:

v 0 m/s (final velocity), u 20 m/s (initial velocity), s 100 m (distance).

Substituting the values into the equation:

0 (20)^2 2{a}(100)

Rearranging for a:

-400 200{a}

{a} frac{-400}{200} -2 text{m/s}^2

The deceleration is -2 text{m/s}^2.

Using Newton's second law, F ma, and assuming that friction is the only resistive force acting on the car, we find:

-{F} 2000 times (-2) text{m/s}^2

{F} 2000 times 2 4000 text{N}

Verification Using the Work-Energy Principle

First, determine the initial kinetic energy based on:

KE frac{1}{2} m v^2 frac{1}{2} times 2000 times (20)^2 400000 text{J}

Work done by the friction force is given by:

W F cdot s

The work equals the loss in kinetic energy of 400000 J:

Rearranging for F:

F frac{W}{s} frac{400000 text{J}}{100 text{m}} 4000 text{N}

Conclusion

The average frictional force that stops the car is 4000 N. This force is acting in the opposite direction of the car's motion and effectively brings it to rest over a distance of 100 meters.