Thread: Varying Acceleration

We know that acceleration is:

a = v1 - v2/t

This is ok if we have a uniform acceleration, but what if the acceleration varies, say from friction or wind resistance on car? Is there a formula, or rather a function for the acceleration given the friction etc, or any other naturally unforseable circumstances that would affect the acceleration?

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Well, that's why they invented calculus.

Recall that a = dv/dt
and F = ma.

So F/m = dv/dt,

in which F is the net force acting on an object. In the case of a falling object, for example, there is gravity opposed by the force of air resistance. But the force of air resistance will vary with v, the velocity. So you must replace F with an expression that includes v. The result might be a rather complex differential equation.

OK so far? Now give us an example of the problem you're working on.
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A vehicle accelerates at 3mph/per second but that changes every second. So the original measurement needs changing every second and it is a very time consuming task, give overall avereaging. I'm just looking for a quick and easy method.

Calculus is quick and easy.

Originally Posted by svwillmer

A vehicle accelerates at 3mph/per second but that changes every second. So the original measurement needs changing every second and it is a very time consuming task, give overall avereaging. I'm just looking for a quick and easy method.

3 mph/sec is a constant acceleration. How does it change with time?

He is thingking of wind resistence. So that would probably mean that acceleration increases uniformly?

Originally Posted by Harold14370

Originally Posted by svwillmer

A vehicle accelerates at 3mph/per second but that changes every second. So the original measurement needs changing every second and it is a very time consuming task, give overall avereaging. I'm just looking for a quick and easy method.

3 mph/sec is a constant acceleration. How does it change with time?

The vehicle is accelerating at an unknown rate per second. The change per unit of time is unknown.

Let's imagine that wind resistance is proportional to the velocity of the whatever. So if the object starts out with initial acceleration a_0 (which is provided by some constant force) and velocity 0 (for sake of ease), then we have:

a = a_0 - kv

where k > 0 is some constant of proportionality. Since a = dv/dt, we have the differential equation:

dv/dt = a_0 - kv

We can solve this for v to find:

v = (a_0 - Ke^(-kt))/k

where K is some constant of integration. This gives us velocity in terms of time, and taking the derivative, we find:

a = Ke^(-kt)

But when t = 0, a = a_0 = K, so we have that a = a_0 e^(-kt). Acceleration decreases exponentially with time. Velocity approaches a limiting value a_0/k.

Thank you

The air resistance will be proportional to the square of the velocity.
http://en.wikipedia.org/wiki/Drag

The standard equation for drag is one half the coefficient of drag multiplied by the fluid density, the cross sectional area of the specified item, and the square of the velocity.

Since you are talking about the acceleration of a car, you might be dealing with a torque-speed curve for an engine that may not be a simple mathematical function. In that case you would have to calculate the thrust of the drive wheels on the road based on the torque-speed curve, the gear ratios, and the tire diameter. You could do a computer iteration perhaps with a spreadsheet program like Excel. Make a lookup table of thrust versus engine speed. You might need several of these for each gear ratio. You would split the time up into say 1 second intervals and calculate the speed at the beginning and end of each interval assuming the acceleration is uniform for that interval. Use the speed at the end of one interval as the initial speed for the next interval. The smaller the interval the more accurate the answer.

My bad! It's been years since I've taken any physics. Thanks for the correction!

The vehicle is accelerating at an unknown rate per second. The change per unit of time is unknown.

That's easy. If everything is unknown then so it is.

The air resistance will be proportional to the square of the velocity.
http://en.wikipedia.org/wiki/Drag
Quote:
The standard equation for drag is one half the coefficient of drag multiplied by the fluid density, the cross sectional area of the specified item, and the square of the velocity.

Only if the drag coefficient is constant. Well, it isn't, it's a function of Reynolds number, Mach number, and other parameters depending on the application.

SteveF essentially got it right in the first reply to the question and there isn't much to be added. You're starting out with an Equation of Motion (Newton's law) and integrate it over time to obtain velocity. However complicated the time history of force (or acceleration) is, it has to be known to have a well-defined problem.

The high school "formula"

v(t2) -v(t1) = a*(t2 -t1)

is for the special case of constant acceleration in which case the "integral of acceleration over time" is expressed by the simple product on the right hand side.

serpicojr provided us with another special case: acceleration as a linear function of velocity (what's a real application for this?).

The general case (that encompasses all special cases) is this:

v(t2) -v(t1) = Integral of a(t) from t=t1 to t2

You need to know acceleration as a function of time, either analytically or by discrete measurements, to perform the integration (analytically or numerically).