Power in 1st Gear

Question

The max power an engine puts out is what determines the max speed a car travel at right, obviously it's more nuanced than that because there are losses in the drive train, and the shape of the car determines it's drag coefficient, but the power output of the engine is the still the top limit and the rest falls out of that.

So here's my question, if I had a manual transmission car and I put it into first gear and let's say the max power occurs at 6000 RPM and the gear ratio in first gear is 3:1 and the final drive ratio is 2:1.

Now let's say I cruise in first gear at the max power output, so my tries are moving at 1000 RPM and for arguments sake lets say that transmits to 40 MPH. And the torque at the tires is 6 times the value it would be at 6000 RPM, I understand that higher gears if I run the engine at max power I get to the top speed the car is capable of and this limit is imposed by the drag power increasing proportionally to velocity cubed, ignoring the losses due to transmission friction and rolling friction and what not.

But in first gear I am going to slow for the power of the drag to be enough to counter that of the engine. But where does the excess power go while in first gear?

There's a lot of torque being produced at the tire and that represent a large force against the pavement and if there's no slipping this is what causes the car to move forward right?

But once we're cruising with the tries at 1000 RPM the force at the tires is still large the engine just can't turn any faster, so what is the offsetting force that causes the car to stop accelerating if it isn't drag?

Is it the engine itself braking the while simultaneously providing the force to keep the car going?

Sorry for rambling, I've been thinking a lot about this question the past couple of days and I have not been able to figure out exactly how these forces are being balanced.

Answer 1

Now let's say I cruise in first gear at the max power output

If you're 'cruising' in first gear at 6,000 rpm, it's unlikely the engine is producing power at anywhere near its maximum.

For rotational motion, power is the product of the $torque$ and the rotational speed.

To produce maximum power, this engine must not only be turning over at 6,000 rpm, it must also be loaded such that this engine will produce the torque required for maximum power output.

For example, imagine this engine is on a test stand and an electronic speed control tries to maintain the rpm at 6,000.

Initially there is only a light load and so this engine is producing a relatively small amount of power. As the load is increased, the power produced by this engine increases even though the rpm is held constant.

There will be some load above which this engine can no longer maintain 6,000 rpm and the power this engine produces with that load is, by stipulation, its maximum power.

So, unless the car is being driven through some kind of 'goop' that just happens to provide just the right wheel resistance such that this engine is loaded for maximum power, you can't conclude by the fact that this engine is turning over at 6,000 rpm that it is in fact producing maximum power.

Answer 2

But where does the excess power go while in first gear?

The engine power is not a pure function of RPM. It is a function of RPM and load. In some regimes (like high RPM in first gear), it is not producing as much power.

As an analogy, think about the power you can exert with your arm. If I give you a baseball, you can throw it and put quite a bit of power into it. $W = F \times d$, so the power your arm is developing depends on the force you exert on the baseball.

Now, I swap the baseball for something tiny like a grain of sand. The sand has such low mass that even at the fastest you can move your arm, its acceleration produces only a very tiny force on your hand. The power you can put into the sand remains tiny.

In your car, if you are accelerating hard in first, the acceleration of the vehicle is creating forces and torques that go back to the engine. It can develop maximum power there.

But if you stop accelerating and just cruise, the torque on the engine is minimal. It can no longer provide maximum power. Because the load is lower, it will require less fuel flow to maintain that RPM as well. (If you kept it floored in first gear, you'd eventually exceed the target RPM).