# Relationship of Power and Driving Force

• Maths_Teacher
In summary: OK, I think the penny has dropped about the necessity of a Gears in order for there to be a variable driving Force, but what about the second point taking the derivative of a fixed Force when it is in fact changing with time either because of a gear change, or perhaps with a continuously variable gearing?The derivative of a fixed force is always zero.
Maths_Teacher
I've been teaching the Work, Energy and Power in UK A'Level Mechanics for some years without a problem. However, I got a question in class today which really made me think about my deeper understanding of the topic. I wonder if anyone can help with explanation of the problems below.

The questions were in relation to problems involving a car being propelled by some kind of engine and experiencing a constant external resistive force.

In this situation the equation Power = Driving Force x Velocity is used. If the power output of the engine is constant then this implies the Driving Force starts at a maximum value and gradually reduces as the velocity increases until the car reachs a maximum velocity and the car stops accelerating. At this pointy the driving force must be equal to the resistive force. When challenged I struggled to explain why the driving force is varying and further why is should start at a high value and then reduce (if anything it seems it should be the other way round). I tried to think in terms of the work done by the engine with some being converted to kinetic energy, as well as heat and sound, but this didn't help.

This all got me thinking about the derivation of P=Dv I’ve always done this by differentiating work (written as force x distance) stating the force is assumed constant. However, in the example above it seems clear that D is some kind of function of t!

I then started thinking about Power in terms of energy transfer and noted that for an idealised system where all work done by an engine is converted to Kinetic Energy, taking the rate of change of KE leads to the same formula (P=DV). So is the above due to some kind of implied modelling assumption?

Can anybody help with a better explanation I can offer my inquisitive sixth formers.

Do you have doubts about force being larger "in the beginning"? Or in lower gears?
The force is larger when you start and decreases as you increase the speed.
Why do you think it should be the other way around?

Not thinking about anything as complex as a gear system. It seemed to me that the engine would need to use more engery a it warms up leaving less to provide ke to the system. For example is the lubrication not more effective as the engine heats? Also does the engining not have to overcome initial ineria when it starts, again leaving little energy to be transferred to ke?

Maths_Teacher said:
In this situation the equation Power = Driving Force x Velocity is used. If the power output of the engine is constant then this implies the Driving Force starts at a maximum value and gradually reduces as the velocity increases until the car reachs a maximum velocity
Maths_Teacher said:
Not thinking about anything as complex as a gear system.
It is precisely the gearing system which allows an engine running at maximum power (and therefore a fixed rotation rate) to operate at a range of velocities. It is precisely the gearing system which results in a reduction in driving force as velocity increases.

The gearing system is not complex. It is simple. Far simpler than competing theories about lubrication efficiency as a function of temperature.

Maths_Teacher said:
If the power output of the engine is constant
Maths_Teacher said:
Not thinking about anything as complex as a gear system.
Without gearing, how do you keep the power constant at different speeds?

nasu said:
Do you have doubts about force being larger "in the beginning"? Or in lower gears?
The force is larger when you start and decreases as you increase the speed.
Why do you think it should be the other way around?
Maths_Teacher said:
It seemed to me that the engine would need to use more engery a it warms up leaving less to provide ke to the system. For example is the lubrication not more effective as the engine heats? Also does the engining not have to overcome initial ineria when it starts, again leaving little energy to be transferred to ke?

OK, I think the penny has dropped about the necessity of a Gears in order for there to be a variable driving Force, but what about the second point taking the derivative of a fixed Force when it is in fact changing with time either because of a gear change, or perhaps with a continuously variable gearing?

Maths_Teacher said:
but what about the second point taking the derivative of a fixed Force when it is in fact changing with time either because of a gear change, or perhaps with a continuously variable gearing?
If force is not constant, then the formula for work is an integral. The power formula doesn't change, except that F is a function of time.

Thanks for that, this makes sense to me now. Great video, I use it my pupils tomorrow.

Note that another contributor that has been glossed over is acceleration. Any force left over after accounting for drag and friction contributes to acceleration via f=ma.

So if you had a constant power, the acceleration starts high and then drops as drag and speed increase.

Here's an excel sheet to look at, its a trial and error algorithm for calculating top speed from predicted engine power and involving other factors such as drag coefficient rolling resistance etc., you can pre calculate the rolling resistance using the rolling resistance tab.
remember the tan boxes are inputs and the v start figure should be greater than zero.
have fun, I've used some example data but you can alter the inputs to suit yourself
i won't be viewing this site again until Monday.

#### Attachments

• top speed example.xlsx
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Thanks for this, a couple of my pupils have interviews for Engineering coming up at Oxbridge, I'll share this with them - just the kind of thing they need as discussion points.

by the way, the maths assumes that the drag co-efficient is the same in both directions.

## 1. What is the relationship between power and driving force?

The relationship between power and driving force is complex and multifaceted. In general terms, power refers to the ability to influence or control others, while driving force refers to the internal or external factors that motivate individuals or groups to take action. The two concepts are closely intertwined, as individuals or groups with power often use it as a driving force to achieve their goals or maintain their dominance. On the other hand, individuals or groups who lack power may use their driving force to challenge and resist those in power.

## 2. How does power impact the driving force of individuals?

Power can have a significant impact on the driving force of individuals. Those in positions of power often have the ability to shape or manipulate the driving force of others, either by rewarding or punishing certain behaviors. This can lead to individuals feeling either motivated or demotivated, depending on how their actions align with those in power. Additionally, individuals with power may use it to suppress or silence the driving force of others, particularly if it threatens their position or authority.

## 3. How does the driving force of a group affect its power dynamics?

The driving force of a group can greatly impact its power dynamics. A group with a strong and united driving force can be a formidable force, as they are more likely to have a clear and shared goal that they are working towards. This can give them a sense of cohesion and strength, potentially leading to an increase in power. On the other hand, a group with a weak or divided driving force may struggle to maintain power, as individual members may have conflicting motivations and goals.

## 4. Can the relationship between power and driving force be positive?

The relationship between power and driving force can be positive in some cases. For example, if those in power use their influence to support and uplift the driving force of others, it can lead to positive outcomes and a healthy power dynamic. Additionally, when the driving force of a group aligns with the values and goals of those in power, it can lead to a more harmonious relationship. However, if power is abused or used to suppress the driving force of others, it can have negative consequences and damage the overall relationship.

## 5. How can individuals or groups maintain a balance between power and driving force?

Maintaining a balance between power and driving force requires open communication and mutual respect. Individuals and groups should strive to understand and support each other's driving forces, while also being mindful of power dynamics and avoiding abuse of power. It is essential to have a shared understanding and clear boundaries in place to prevent one side from becoming too dominant. Regular reflection and evaluation of power dynamics can also help ensure a healthy balance between power and driving force.

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