At a particular instance, a train of mass 147 metric tons was accelerating along a horizontal section of track at 68 centimeters per second squared against the resistive force of 50 kilogram-weight for each metric ton of its mass. Given that the maximum speed of the train along this section of track was 72 kilometers per hour, find the power of its engine.
We begin by noticing that this question gives us several values in nonstandard units. We will therefore begin by converting these into standard units. There are 100 kilograms in one metric ton. This means that the mass of the train is equal to 147 multiplied by 1000 kilograms, which is equal to 147000 kilograms. There are 100 centimeters in one meter. This means that in order to convert the acceleration from centimeters per second squared to meters per second squared, we need to divide 68 by 100. The acceleration of the train is 0.68 meters per second squared.
Next, we recall that one kilogram-weight is equal to 9.8 newtons. This means that the resistive force is equal to 50 multiplied by 9.8 newtons for each metric ton of its mass. Since the mass of the train was 147 metric tons, we need to multiply 50 by 9.8 by 147. The resistive force at this point is therefore equal to 72030 newtons.
Finally, since there are 1000 meters in one kilometer and 3600 seconds in one hour, we can convert a speed of 72 kilometers per hour into meters per second by multiplying by 1000 and then dividing by 3600. This is the same as dividing by 3.6, giving us a maximum speed of 20 meters per second. We will now sketch a diagram to model the situation.
The train has a mass of 147000 kilograms. It is accelerating at 0.68 meters per second squared. And there is a resistive force of 72030 newtons. We will let the driving force of the train be 𝐹 newtons. And we know that we can calculate the power of the engine by multiplying this force 𝐹 by the velocity of the train 𝑣. In this question, we’ve already calculated that the maximum speed or velocity is 20 meters per second.
Recalling Newton’s second law 𝐹 equals 𝑚𝑎, we know that the sum of the vector forces is equal to the mass multiplied by acceleration. If we let the positive direction be the direction of travel, we see that the sum of the forces is equal to 𝐹 minus 72030. This is equal to the mass of 147000 kilograms multiplied by the acceleration of 0.68 meters per second squared. The right-hand side simplifies to 99960. We can then add 72030 to both sides of our equation such that 𝐹 is equal to 171990. The driving force of the train is 171990 newtons.
The power 𝑃 in the standard unit of watts is therefore equal to 171990 multiplied by 20, which is equal to 3439800 watts. Whilst we could leave our answer in these units, in this question we will convert our answer to horsepower. We recall that one horsepower is equal to 735 watts. This means that we can calculate the power in horsepower by dividing 3439800 by 735. This is equal to 4680. The power of the train’s engine is 4680 horsepower.