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Mr. Kovalcin,

I teach AP physics at The Woodlands High School in Texas. I made up a problem in which I give the students a simple equation for velocity as a function of time: v(t) = 4 + 3t^2 and the mass of the object exhibiting this motion (10 kg). I then ask the students to determine multiple values from this function, including the work done (delta KE) from t=0 to t=4 and the impulse (delta p) from t=0 to t=4.

I run into a problem, though, when I tried to calculate the average force over that time interval. Using integration, I could find the displacement of the object for that time interval. Thus, I thought I could find average force using two methods:

1. Divide work done by total displacement

2. Divide impulse by total time

and that these two calculations would yield the same results. They do not, however.

Which method, if either, is accurate? Where is my flaw in the incorrect method?

Thank you.

The flaw in your analysis is that the resulting force is not linearly proportional to time. As a result the average does not lie midway between the end points.

As an example I would like to use a somewhat simpler velocity function to reduce the mathematical complexity - this simplification will not change the outcome.

I am going to assume that the velocity function is : v(t)=3*t^2

Taking the anti derivative the displacement function will be:

D(t)=3*t^3/3=t^3 (assuming the displacement is 0 at t=0)

Taking the derivative of the velocity function the acceleration function,will be:

a(t)=6*t

And the force function will be:

F(t)=m*a=10*6*t=60*t

Evaluating each of the these functions at t=4s:

a=6*4=24m/s^2

v=3*4^2=48m/s

d=t^3=4^3=64m

F=60*4=240

Calculating the work done is given by:

W=Integral[F dot dx]

In this case dx can be determined from the displacement function from above:

x=t^3 which becomes dx=3*t^2dt

So the integral for the work done becomes:

W(4) = Integral [F dot 3*t^2 dt] = Integral [60*t*3*t^2 dt] = Integral [180*t^3 dt] = 180/4*t^4 (4) = 45*4^4 = 11520J

Average force would then be given by:

Fave = W/d = 11520J/64m = 180N

Since the average force developed from the initial and final forces (at 0s and 4s) will be:

Fave = (0+240)/2 = 120N

Why are these answers different? Because the force as a function of time is non-linear!

F(t) = 180*t^3

And the average of the function, since it is a curve, does not lie on the midpoint of the curve.

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I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.