A Train Engine Pulls Two Identical Cars Behind It

Introduction

A train engine is a locomotive that pulls a train of cars. Train engines are typically powered by diesel fuel or electricity. The engine provides the power to move the train forward, and the cars provide space for passengers or freight.

In this problem, we consider a train engine that is pulling two identical cars behind it. The engine has a mass of M, and each car has a mass of m. The train is moving at a constant speed of v.

Problem Statement

What is the force of friction between the engine and the track?

Solution

The force of friction between the engine and the track is equal to the force required to overcome the resistance of the air and the rolling resistance of the wheels. The resistance of the air is proportional to the square of the velocity, and the rolling resistance is proportional to the weight of the train.

In this problem, the resistance of the air is given by:

F_air = 0.5 * rho * A * v^2

where:

* rho is the density of air
* A is the frontal area of the train
* v is the velocity of the train

The rolling resistance is given by:

F_roll = f * m * g

where:

* f is the coefficient of rolling resistance
* m is the mass of the train
* g is the acceleration due to gravity

The total force of resistance is:

F_res = F_air + F_roll

The force of friction between the engine and the track is equal to the force required to overcome the total force of resistance. Therefore, the force of friction is:

F_f = F_res

Substituting the expressions for F_air, F_roll, and F_res into the expression for F_f, we get:

F_f = 0.5 * rho * A * v^2 + f * m * g

This is the force of friction between the engine and the track.

Example

Consider a train engine with a mass of M = 100,000 kg and two identical cars with a mass of m = 50,000 kg each. The train is moving at a constant speed of v = 100 km/h. The density of air is rho = 1.2 kg/m^3, the frontal area of the train is A = 10 m^2, and the coefficient of rolling resistance is f = 0.005.

The force of friction between the engine and the track is:

F_f = 0.5 * 1.2 * 10 * 100^2 + 0.005 * 200000 * 9.81

F_f = 120000 N

Conclusion

The force of friction between a train engine and the track is equal to the force required to overcome the resistance of the air and the rolling resistance of the wheels. The resistance of the air is proportional to the square of the velocity, and the rolling resistance is proportional to the weight of the train.