Prof. N.H. CHEUNG
Department of Physics, Hong Kong Baptist University

The train moves thousands of passengers at a hundred km per hour, making frequent stops along the way. What kind of acceleration is involved? How powerful is the engine? How much electricity is consumed? By using simple Newtonian mechanics, we can answer these questions quantitatively.

You can experience Newtonian mechanics while traveling by train. Imagine going from Tai Wo to Fanling, a scenic ride of about 6 km. Let's idealize this section of the track to be straight and leveled. As the train leaves the Tai Wo station, it accelerates from rest to a final cruising speed of 100 km/hr over a distance of about 1 km. We can model the velocity v as a function of distance traveled x by an exponential function; as shown in Fig. 1. Though simple mathematically, v(x) is realistic enough for most estimates, as we will show.

(The route of train from Tai Wo to Fanling)

(Fig. 1. The train gradually picks up speed as it leaves the Tai Wo station for Fanling. The dependence of the train velocity v on distance traveled x may be modeled by, where is 100 km/hr, is 2.5 , and x is in km.)

(Fig. 2. Based on v(x) given in Fig. 1, the train acceleration a(x) as a function of x can be plotted (red curve). The pulling force can be plotted as well (right-hand axis). Two cases are shown: when the resistance is neglected (red curve) and when it is included (blue curve).)

We will first calculate the acceleration a(x) as a function of x. Using elementary calculus, we have,

; (1)

so a(x) is simply the slope of the curve of v(x) vs x (Fig. 1) times the velocity. This is plotted in Fig. 2 (red curve, left-hand axis). As can be seen, the train accelerates maximally near the beginning, and soon travels at a constant velocity. The peak acceleration is about 5% of the acceleration due to gravity, which is well within the comfort zone of most people.

We now estimate the force exerted by the engine to accelerate the train. There are typically 12 carriages per train, each weighing about 40 metric ton when empty. Loading capacity is about 300 passengers each. So the total mass is about kg. The start up force F(x) is simply the mass times the acceleration, which is depicted in Fig. 2 (red curve, right-hand axis). Notice the start up force can be as high as 340 kN. Two observations should be drawn. First, a train normally has four engines. Each can exert a maximum pulling force of about 100 kN. So the train is working just about as hard as it possibly can at start up. Second, the coefficient of friction between the wheels and the railroad track is about 0.1 when cruising and increases to about 0.3 when stationary. Given a mass of kg, the frictional drag (700 - 2,000 kN) is certainly large enough to prevent slippage.

The work done dW by the engine is F dx. The mechanical power P expended is dW/dt = F dx/dt = F v. This is plotted in Fig. 3 (red curve), as a function of x. The maximum power drawn is about 5.5 MW, or about 7,400 horsepower. This is reasonable because we are transporting the mass equivalent of about people. Requiring 7,400 horses sounds about right.

(Fig. 3. The power expended P as a function of x. Two cases are shown: when resistance is neglected (red curve) and when it is included (blue curve).)

According to the red curve in Fig. 3, the engine can be turned off once the train has reached its cruising speed of 100 km/hr. This is certainly unrealistic. Even when the train is not accelerating, the engine still has to work hard to overcome all kinds of mechanical friction, including bearing resistance of the moving parts and wheel-track resistance. This mechanical resistance is found to increase with train velocity. For local trains, (in N) may be modeled by a polynomial in v (in km/hr), . In addition to mechanical friction, there is air drag as well. For still air, the aerodynamic resistance (in N) varies approximately as the square of the train velocity v (in km/hr), . The total resistance R is simply the sum of and ,

. (2)

Therefore, the total force exerted by the engine is the sum of R and the start-up force F(x). is the blue curve shown in Fig. 2 (right-hand axis). As can be seen, force against resistance is a minor component near start up, but becomes the sole effort while cruising. Similarly, more power is expended when resistance is taken into account, as depicted by the blue curve in Fig. 3.

What is the cost of electricity to power this leg of the journey? Let's calculate the total work done. It is given by , which is the area under the blue curve in Fig. 2, or about J. Assuming an efficiency of about 70 %, the total electrical energy consumed is about J. The cost of electricity (high power tariff) is about $0.50 per kW-hr, so the electricity bill comes to about $125.

(A sample of electricity bill)

Hopefully, you now appreciate how physics can help you estimate, quantitatively, the key dynamical parameters of train motion based on just a few assumptions. When you find the carriage lights dim or the air-conditioning weakens as the train starts off at Tai Woo, you know the engine is pooling power to accelerate the massive vehicle. And you know what percent of your train fare goes to paying the electricity bill. Of course, you will find a lot more examples of Newtonian mechanics around you. So by the time when you become the first astronaut from Hong Kong, you know deep down that space travel is just one more application.

In the mean time, please relax and enjoy the rest of the scenic ride.