How do we find the uncertainty of v in the kinematics eqn?

In summary, the homework statement is that given the kinematics equation v2 = u2 + 2as, find the uncertainty Δv. The attempt at a solution is to apply the rules of uncertainty propagation. The steps involved are to apply the addition rule, the product rule, and the bc2 rule to calculate the uncertainty Δ(2as).
  • #1
minamikaze
9
2

Homework Statement


Given the kinematics equation v2 = u2 + 2as, find the uncertainty Δv.
Given: Values of a, s, u, and their associated uncertainties.

Homework Equations


v2 = u2 + 2as

The Attempt at a Solution


I'm aware of the rules for uncertainty propagation, but what do I do in this case where there are powers, sums and products all in one equation?
 
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  • #2
Just combine the rules of uncertainty propagation. What do you know about the rules for the uncertainty of sum and the uncertainty of product?
 
  • #3
I know that if a = b + c, then Δa = Δb + Δc.
And if a = bc2, then Δa/a = Δb/b + 2Δc/c. But I don't understand how the two can come together in the case of v2 = u2 + 2as.
 
  • #4
minamikaze said:
I know that if a = b + c, then Δa = Δb + Δc.
And if a = bc2, then Δa/a = Δb/b + 2Δc/c. But I don't understand how the two can come together in the case of v2 = u2 + 2as.
Just apply your two rules sequentially. v2 = u2 + 2as has the overall form of a = b + c, where a is v2 etc.
 
  • #5
So, (Δv)2 = (Δu)2 + 2ΔaΔs ?
 
  • #6
minamikaze said:
So, (Δv)2 = (Δu)2 + 2ΔaΔs ?
No, I wrote "where a is v2 etc." If a represents v2 what is Δa?
 
  • #7
Δv2 ? What is Δv2 and how do you calculate it?
 
  • #8
minamikaze said:
Δv2 ? What is Δv2 and how do you calculate it?
Use your bc2 rule.
 
  • #9
I'm sorry, I already said I know the rules and have listed them down. This isn't really leading me anywhere.
Let me phrase my question very, very clearly.

I know that for a = bc2, Δa/a = Δb/b + 2Δc/c.
What about a = b+c2, where there is both a product and a sum, and a square? I just do not see how this is coming together - the sum propagation and the product propagation rule.
It cannot be the same, Δa/a = Δb/b + 2Δc/c, isn't it?

So you advised applying the rules sequentially.
Step 1 : Apply the addition rule.
Δa = Δb + Δc2
Step 2: Apply the product rule.
Δa/a = Δb/b + 2Δc2/c

Is this it?

And so for my original question, should it be:
2Δv/v = 2Δu/u + 2ΔaΔs ?
It does not seem right, and I would like to get clear guidance on this, instead of just asking me to apply this rule and that rule, which I already know, and have stated clearly above.
 
Last edited:
  • #10
Yes we have to apply the sum and product rules however it is needed, but it seems you doing some mistakes applying the product rule.

By applying the sum rule we get

##\Delta v^2=\Delta u^2+\Delta(2as)##

Now we going to apply the ##bc^2## rule for the first 2 terms of the sum (two different applications of the rule).
So we have applying the ##a=bc^2## rule for ##a=v^2 ,b=1, c=v##
##\frac{\Delta v^2}{v^2}=\frac{\Delta 1}{1}+2\frac{\Delta v}{v}\Rightarrow \Delta v^2=v^2(0+2\frac{\Delta v}{v})\Rightarrow \Delta v^2=2v\Delta v##

Similarly applying the rule for ##a=u^2,b=1,c=u## we get ##\Delta u^2=2u\Delta u##

Now what rule should we apply to calculate ##\Delta (2as)##. We don't have a squared term here to apply the bc^2 rule and what you did is that
##\Delta (2as)=2\Delta a \Delta s## but this is not correct is it?
 
  • #11
z = 2as
Δz/z = Δa/a + Δs/s
∆z = z(Δa/a + Δs/s)

Is this it?
 
  • #12
So overall, it would be:

2vΔv = 2uΔu + 2as(Δa/a + Δs/s)

Is this correct?
 
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  • #13
minamikaze said:
So overall, it would be:

2vΔv = 2uΔu + 2as(Δa/a + Δs/s)

Is this correct?
Yes, but you can simplify that with some cancellation.
 
  • #14
Thank you, I am very clear about it now.
The end result will be:
2vΔv=2uΔu + 2sΔa + 2aΔs

Thanks all for the help.
 
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What is the formula for calculating uncertainty in velocity?

The formula for calculating uncertainty in velocity is ∆v = ∆d/∆t, where ∆v is the uncertainty in velocity, ∆d is the uncertainty in distance, and ∆t is the uncertainty in time.

What is the significance of finding uncertainty in velocity?

Finding uncertainty in velocity is important because it helps us understand the accuracy of our measurements and the potential errors in our calculations. It also allows us to make more precise predictions and draw more accurate conclusions.

How do we determine the uncertainty in distance and time?

The uncertainty in distance can be determined by measuring the smallest division on the measuring instrument used and dividing it by 2. The uncertainty in time can be determined by dividing the smallest division on the stopwatch by 2. It is important to always use the same units when determining uncertainties.

What factors can contribute to uncertainty in velocity?

Factors that can contribute to uncertainty in velocity include limitations of the measuring instrument, human error in taking measurements, and external factors such as wind or friction affecting the motion of the object.

How can we decrease the uncertainty in velocity?

To decrease the uncertainty in velocity, we can use more precise measuring instruments, take multiple measurements and calculate the average, and minimize external factors that can affect the motion of the object. We can also improve our technique and accuracy in taking measurements.

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