Weighted Average Velocity between r1 and r2

  • Thread starter Thread starter natski
  • Start date Start date
  • Tags Tags
    Average
AI Thread Summary
The discussion focuses on calculating the average velocity of a body moving between two positions, r1 and r2, when the velocity is a function of position rather than time. The initial formula proposed, using integrals of velocity and position, is deemed insufficient for finding the average velocity. The conversation highlights the need to integrate over time, leading to the conclusion that time can be expressed as a function of position, allowing for a valid average velocity calculation. The participants clarify that while integrating over position may seem valid, it does not yield the correct average velocity unless expressed in terms of time. Ultimately, the correct formulation for average velocity is confirmed as vbar = Integral[dr] / Integral[dr/v].
natski
Messages
262
Reaction score
2
Consider a velocity which is a function of position r, which does not vary linearly with time.

Consider a body moving with this varying velocity between distance r1 to r2.

Let us define the average velocity between r1 and r2 as (r2-r1)/time taken to travel between r2 and r1.

I assumed the average would be found by:

Integral[ v(r) dr {r2, r1}] / Integral [dr {r2, r1}]

But this formula does not seem to work. Are there any special cases where this formula is not sufficient?
 
Last edited:
Physics news on Phys.org
You need the integral over time to get the average velocity.
 
But time is a function of position, so there does not seem any reason why one could not use the position as an equivalent weight?

So you suggest:

Integral[v dt] / Integral[dt]

But
r(t) => dr/dt = 1/f'
=> dt = f' dr

Hence one could write Integral[v f' dr] / Integral[f' dr]

But dr/dt = v
hence f' = 1/v
=> vbar = Integral[dr] / Integral[dr /v]
 
Last edited:
natski said:
So you suggest:

Integral[v dt] / Integral[dt] ...(1)

...

But dr/dt = v
hence f' = 1/v
=> vbar = Integral[dr] / Integral[dr /v] ...(2)

The two integrals (1) and (2) are identical. Sorry, I don't get your point now...?
 
Well Meir felt you couldn't take the weighted average by integrating over position, but I am trying to prove that you can by virtue of the fact that his suggestion, of integrating over time, can be written as an integral over position.
 
natski said:
Well Meir felt you couldn't take the weighted average by integrating over position, but I am trying to prove that you can by virtue of the fact that his suggestion, of integrating over time, can be written as an integral over position.

Ok. But Meir was responding to this in your original post:

"Integral[ v(r) dr {r2, r1}] / Integral [dr {r2, r1}]"

which is not going to give average velocity.

But "vbar = Integral[dr] / Integral[dr /v]" in your 3rd post will give average velocity.
 
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
Back
Top