Average Acceleration of a Accel vs Speed Graph

In summary: In general, the distance traveled is equal to the integral of (v/a)dv, where v is the instantaneous velocity and a is the instantaneous acceleration. In the case of constant acceleration, this simplifies to the well-known formula Distance = (Vfinale^2)/(2*Acceleration).
  • #1
lboucher
2
0
Hi All

So I have an Accel vs Speed graph and data.
Goes from 0 to 80 mph.
This is all the info I have.

I would like to figure out what Average Acceleration would give me the same distance, of an acceleration from 0 MPH to 80 MPH.
AKA, using the formula Distance = (Vfinale^2)/(2*Acceleration)

Any ideas on how to figure this out?
 
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  • #2
If you plot 1/acceleration versus speed, your function value is proportional to the time spent at a specific velocity (assuming your acceleration is always positive).
If you plot speed/acceleration versus speed, the area below the function is proportional to the distance travelled.
 
  • #3
Sorry, but that still confuses me as the units don't add up. Also when I attempt to plot this out in excel and add up the area under the curve, the resulting numbers cannot be right
 
  • #4
The area under the v/t graph should give total distance, whatever the instantaneous velocity and accelerations are. Does that hold for an (a/v)/v graph too? Is there not a constant needed after the integration?
 
  • #5
A simple example: v(t)=at+c with constant a and c.
Distance after time T is ##\frac{1}{2}aT^2 + cT##.

v/a = t + c/a
t=(v-c)/a
Therefore, v/a = (v-c)/a + c/a = v/a (trivial in this example, as a is constant)
Speed increases from c to c+aT. If we integrate v/a from c to c+aT, we get ##\int \frac{v}{a}dv = \frac{1}{2a}((c+aT)^2-c^2) = cT+\frac{1}{2}aT^2## - the same as above.
 
  • #6
Wine and gin make this less accessible but the sums seem right in this case. It that also OK for non constant acceleration? I guess it could be - piecewise. Good one. I love it when someone else does the algebra and I can follow it.
 
  • #7
mfb said:
If you plot 1/acceleration versus speed, your function value is proportional to the time spent at a specific velocity (assuming your acceleration is always positive).
If you plot speed/acceleration versus speed, the area below the function is proportional to the distance travelled.
Not just proportional to the distance traveled. It is equal to the distance traveled. Another way to do this is to plot 1/(2a) as a function of v2. This will also be equal to the distance traveled.
 
  • #8
I did not check the prefactor when I wrote my first post here, but it is 1, right.

It that also OK for non constant acceleration?
It works there as well, but the integration can get more difficult.
 

FAQ: Average Acceleration of a Accel vs Speed Graph

1. What is the definition of average acceleration?

The average acceleration is the rate at which an object changes its velocity over a certain period of time.

2. How is average acceleration calculated?

Average acceleration is calculated by dividing the change in velocity by the change in time.

3. How is the average acceleration represented on an accel vs speed graph?

The average acceleration is represented by the slope of the line on an accel vs speed graph. A steeper slope indicates a higher average acceleration.

4. How does the shape of an accel vs speed graph affect the average acceleration?

The shape of the graph does not affect the average acceleration. As long as the slope remains constant, the average acceleration will be the same.

5. What is the relationship between average acceleration and speed on an accel vs speed graph?

The average acceleration and speed have an inverse relationship on an accel vs speed graph. This means that as the average acceleration increases, the speed also increases, and vice versa.

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