**Discussion on Determining an Approximation of the Vehicle's Power Output**

At this stage of the project, the only data that is available is the RPM of the engine, as well as the vehicle's speed.

Knowing the vehicle's speed, both the distance the vehicle travels over a given duration of time as well as the acceleration of the vehicle over a given duration of time can be calculated. Recalling some rudimentary physics concepts, for a given velocity v, and an arbitrary duration of time t, both distance and acceleration can be derived with the following formulas:

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distance=velocity*time
acceleration=velocity/time
*

These formulas are exact provided the velocity of the object is constant over the given duration of time (and in the real world, this is seldom the case). However, both of these formulas will help in finding a fairly accurate approximation for power output.

**Power:**

So how is power output found? Some excellent background material (including unit conversions) can be found here. Power is defined as the rate of using energy, and the typical unit of measurement is called a Watt. A Watt is a Joule/second. Power can be determined with the following formula:

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PowerOutput=EnergyExpended/TimeDuration
*

**Energy:**

A Joule is a unit of energy. Energy is defined as the capacity for doing work. A Joule is a Nm (Newton-meter). For the purposes of this project, it is assumed that the energy available is essentially infinite (as long as there is always gasoline in the tank that is), but we are concerned with the amount of energy expended within a given amount of time that accomplishes the work done. The energy expended will be given by the work done (which is also measured in Joules). The work done can be determined with the following rudimentary formula (since work and energy have the same unit of measure, and the energy available is infinite, we consider the amount of energy expended):

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EnergyExpended=Force*Distance
*

**Force:**

What is Force? Force is measured in Newtons. A Newton is a kilogram*meter/second^{2}. It is defined as an influence which changes the motion of an object. For the project, we will be concerned with finding the Force responsible for changing the motion of the car. We can determine the Force responsible for changing the motion of the vehicle with the following formula:

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Force=Mass*Acceleration
*

**Combining the Concepts:**

We know have all the formulas necessary to make our calculations. So we use them in the following manner:

The mass of the vehicle (in kilograms, units are important) is known. The acceleration of the vehicle over an arbitrary duration can be determined by analyzing the speed of the vehicle over time. Since Force=Mass*Acceleration, we now know the Force applied to the vehicle in the direction of motion we are concerned with (namely forward).

Now that the force is known, we use this value combined with the distance travelled within the same duration of time (which we can get from analyzing the velocity over time). These two values allow us to solve the formula EnergyExpended=Force*Distance.

Now that the energy expended is known, we use it and the duration of time to solve the formula PowerOutput=EnergyExpended/TimeDuration to determine power output in Watts. Conversion from Watts to Horsepower (the conventional unit when dealing with automotive power output) is simple: 1 HP = 746 Watts.

**Sources of Error:**

This theoretical model to determine the power output has its inaccuracies. They are:

- The engine output is not the only force acting on the vehicle. Other frictional forces (such as wind resistance) may act in a direction opposite to the forward motion of the vehicle and are not considered. There are other, more subtle frictional forces at work also (for example the friction on the wheel bearings when the wheels rotate).
- Non-linear changes in variables over time durations. Consider the process of determining the acceleration of the vehicle within an arbitrary amount of time. Although the initial and final velocities are known, the acceleration within the duration of time may not be constant. The linear formula to calculate acceleration does not account for this - it is accurate only when the acceleration is constant over the time duration. This can adversely affect the accuracy of any formulas the value influences.
- The vehicle's mass. Dependent on the number of passengers seated, the cargo, the amount of fluids in the various systems, etc., will influence the mass of the vehicle. Therefore, relying on the curb weight of the vehicle alone may be an inaccurate estimate of the true mass of the vehicle during operation at any instance.