Why does the acceleration of a car decrease as the speed increases?

- anonymous

Why does the acceleration of a car decrease as the speed increases?

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- saifoo.khan

The engine has to do more work.

- anonymous

aha.. but the force that it puts keeps on decreasing right?

- anonymous

so its because ultimately its the POWER that an engine can deliver is a CONSTANT right?.. not force or work

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- hba

Say you are starting up from a stop light in your car (for simplicity assume you have an automatic transmission). If you push the gas pedal part way down and hold it in the same position, then the car will start out at its maximum acceleration. The faster the car goes, the less the acceleration will be.

- anonymous

well.. the original question is WHY that happens :P.. i know what you said happens.

- hba

u can also feel it urself

- hba

its a normal condition u see

- hba

Suppose you are moving at 5m/s at t=0. Acceleration is 5m/s2 initially. At t = 1 your velocity will be 10m/s. If acceleration became 3m/s2, then your velocity at t = 2 will be 13m/s. Thus your velocity still increased but comparatively lesser in amount. Now if acceleration became -8m/s2, velocity at t = 3 will be 5m/s, thus velocity decrease due to negative acceleration.

- anonymous

You might like to consider the air resistance acting on the car - as the speed increases the power required for a given acceleration increases. Similarly, the fuel consumption increases exponentially as speed increases which shows that the air resistance increases greatly with speed. The power of the engine might increase somewhat as engine speed increases but the increase is not enough to overcome the increasing air resistance so that the acceleration decreases.

- anonymous

ebaxter has nailed it: the principal reason is air resistance, the force of which rises as the third or fourth power of the velocity. As the car goes faster, the force of the engine is opposed by a greater and greater force of air resistance, so the net force -- and the acceleration -- gets smaller and smaller, until at the maximum speed the engine's force is just sufficient to counter air resistance, and then the acceleration falls to zero.
This is even true for an "engine" like the force of gravity which has no limits on the amount of power it can provide. That is why falling objects, if they fall long enough, stop accelerating and reach what's called their terminal velocity -- a steady speed beyond which they do not accelerate. This is the speed at which the force of gravity is countered by the force of air resistance, and there is no net force.
For human beings terminal velocity is pretty low, I think about 180 MPH, but for heavy and streamlined objects like aircraft it can be much, much higher -- thousands of miles per hour. So as a matter of fact, if you are in an airplane headed straight down (admittedly a strange situation, but let's say the wings have fallen off) you would be better off jumping out of the airplane, to take advantage of your personal lower terminal velocity.

- anonymous

however.. if we were to neglect the force of friction (air and ground).. still the engine would finally max out cause it has a DEFINITE power at which it can work and therefore acceleration would be zero ultimately right?

- anonymous

if i am wrong.. i want to know WHAT about the engine is constant.. i am assuming the POWER it can deliver is a constant!

- anonymous

Which friction? If you neglect the friction of the car with the air and road, then you still have the internal friction of the engine and drivetrain itself -- the friction between the pistons and the cylinders, in the bearings, et cetera, and this, too, will increase with engine velocity, so there is a point at which all the engine power would be going to overcoming internal friction. But I suspect long before that point the engine would be damaged by the extreme heat generated by that much energy being dissipated in the bearing and cylinder surfaces through friction.
If you ignore friction within the engine, too, then there is still an upper limit in a single gear because the engine cannot run fast enough -- the gasoline vapor and air can't get into or out of the cylinder fast enough, the centrifugal force on the moving parts starts to damage them, et cetera.
But on the other hand, if you allow yourself an infinite number of gears in your transmission, as well as ignoring all friction forces, internal and external, then I see no upper limit (other than the speed of light) for the velocity to which your engine can accelerate your car. Even the smallest engine, and even the largest car.
If you have constant power P = dEk/dt, where Ek = kinetic energy, then for plain Newtonian mechanics the derivative on the RHS is m v dv/dt. Multiplying through by dt and Integrating both sides gives v^2 = 2 P t / m plus some constant, or
\[v(t) = v(0) + \sqrt{\frac{2 P}{m}} t^{1/2}\]
which can be differentiated once to give the acceleration:
\[a(t) = \sqrt{\frac{P}{2 m}} t^{-1/2}\]
It's interesting that the velocity grows more and more slowly, and the acceleration slows down with time. You're right it asymptotically approaches zero, but not before both time and velocity both reach infinity.
Of course, to do this right we have to use relativistic kinematics, and I never touch relativity before lunch.

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