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I'm trying to buy a car that really feels fun to drive, and the one I test drove doesn't feel all that fun despite having high horsepower, torque, and acceleration. I'm trying to find a mathematical method for observing the car's push-you-back-in-your-seat factor. It's not just acceleration.

http://www.city-data.com/forum/automotive/1977660-how-get-pinned-pushed-into-seat.html You can't just say horsepower or torque. Some cars with less horsepower and torque have given the push you back into your seat feeling more than cars with higher. It's about the rate of change in these. Some say the more linear the horsepower over RPM graph the less push you feel. Does that mean it's the second or first derivative of Horsepower/RPM? Rate of change of a curve sounds like second derivative. . d^2HP/d^2RPM. Does that have a certain term? Power = torque / angular speed, and in this case horsepower = torque / RPM. Therefore d^2(HP)/d^2(RPM) = d^2(torque*RPM)/d^2(RPM). Can this be simplified any further? Is this a correct interpretation of the sudden feeling of getting pushed back, and if so is there a term or some quantity that helps describe it instead of just the second derivative of HP/RPM?

And perhaps it's the rate of change of Torque/RPM instead of HP/RPM.. I'm not sure.

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  • $\begingroup$ What makes you say it's not just acceleration? Naively, the larger the acceleration, the more normal force you will feel from the seat back. If you can help explain why acceleration doesn't capture the effect you're talking about, it could help us determine exactly what does capture it. $\endgroup$
    – d_b
    Commented Sep 30, 2020 at 23:04
  • $\begingroup$ For example, the car I drove has a 0-60 acceleration of around 5 seconds. Yet it didn't push me back in the seat like say a car with a less linear power curve would. To be specific a 2010 Ford Sho supposedly pushes you back in your seat way more than a Lexus IS 350, yet they both have a 0-60 of 5 seconds. Explanations for this include more low-end (low RPM) torque from the Ford SHO, and that the Lexus has a relatively linear Power Curve (power over RPM). $\endgroup$ Commented Sep 30, 2020 at 23:30
  • $\begingroup$ To comment further, most Lexus' don't "feel" fast despite having high acceleration. I guess I'm leaning towards the fact the 2nd derivative of power by time is 0 or approaching 0, making you not "feel" it even though perhaps the force is greater. So maybe my answer is that as the 2nd derivative of power by time approaches 0 the less of a push you feel in a car, and the converse is also true. $\endgroup$ Commented Sep 30, 2020 at 23:32
  • $\begingroup$ Check out these Tesla videos: youtube.com/watch?v=y1bYcetcufw and youtube.com/watch?v=I5mZ1IK7SuU $\endgroup$
    – Bill N
    Commented Oct 1, 2020 at 1:22

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From a physics point of view, it's exactly acceleration. The larger the acceleration of the vehicle, the greater the force the seat is pushing on you to achieve it. (You could of course render acceleration as a function of mass and torque or power, but that's just complicating it)

That said, our bodies are not precision instruments, and our perception of "force" is not perfect. A smaller, louder, windier car may "feel speedier" than a big, quiet boat, even if the actual acceleration is smaller. But those perceptions are not easy to tie an equation to.

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  • $\begingroup$ The motion-related quantity is indeed simply acceleration. I'll just add seat design to the other factors BowlofRed has suggested in the second paragraph above. $\endgroup$ Commented Sep 30, 2020 at 23:23
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There's acceleration and then there's also "jerk." Jerk is the rate at which the acceleration changes i.e. da/dt. A car that goes, say, 0 to 60 in four seconds can do so in many different ways such as a constant acceleration throughout or a big jump in acceleration at the beginning. So you're right that it's not just acceleration. It's also the rate of change of acceleration. I don't know how that relates to the engine characteristics.

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The average acceleration can the same from 0-5s, but if it is not constant, there are necessarily moments of higher acceleration, giving a bigger push-you-back-in-your-seat feeling.

But of course the feeling itself is caused by acceleration.

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The seat and how it responds to acceleration is a big factor too.

A seat with a geometry designed to hug the driver and wrap around sides will give a firm feeling of authority anf can damp undesired shakes and accelerations.

A seat with a larger contact area in the back and shoulder area will impart less force at the same acceleration than a seat with smaller contact area.

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I think Throttle response is what you are looking for. Then again I have a very basic highschool understanding of physics so bear with me

Since what you are feeling, is a result of jerk, basically the increase of the pseudo force counteracting your acceleration, you can directly look at what the engine does under throttle, since it is its direct cause.

Look at a traditional power curve in relation to rpm chart, and imagine your revs going up with your throttle input, as a vertical line moving horizontally through the graph.

Since power is a force over a distance in an amount of time, holding a steady rev count in a flat spot on the graph, will mean no further acceleration, the amount of work output per rev doesn't increase, you are in a state of inertia once again, and you feel bored. You hence decide to put your foot down, and your engine gets into a spot where power rises either gently by 20 horsepower over 500 revs or harshly by 20 horsepower by one revolution. You accelerate for the time it takes your engine to spin up 500 revs or 1 rev. Even if counter intuitive, the time and resulting acceleration in these scenarios can be equal.

Now,if we assume no gear changes happen, the shape of our horsepower curve can show us our flat out acceleration. If we think back, jerk, is the rate of change of acceleration, on our graph acceleration is modeled by being the inclines on our power curve, so jerk is just how fast our imaginary line crosses an incline, or in effect, how fast our imaginary line goes from 0 to redline with your foot pinned to the floor, which is what is what throttle response is!

Throttle responsiveness naturally increases with lighter flywheels, lighter internals and lessening friction between engine components.

Unfortunately you will rarely find figures for specific cars unless it's a marketing party trick like for the Lexus lfa or gma t.50 so I'd just look for car reviews that point out fast throttle response and go from there.

This is really all over the place but I had a similar question and thought it might be valuable to post my conclusion in here, hope it helps! (and is accurate).

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