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

## Homework Statement

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

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?

Delta2
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Just combine the rules of uncertainty propagation. What do you know about the rules for the uncertainty of sum and the uncertainty of product?

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.

haruspex
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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.

So, (Δv)2 = (Δu)2 + 2ΔaΔs ?

haruspex
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So, (Δv)2 = (Δu)2 + 2ΔaΔs ?
No, I wrote "where a is v2 etc." If a represents v2 what is Δa?

Δv2 ? What is Δv2 and how do you calculate it?

haruspex
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Δv2 ? What is Δv2 and how do you calculate it?

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:
Delta2
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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 gonna 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?

z = 2as
Δz/z = Δa/a + Δs/s
∆z = z(Δa/a + Δs/s)

Is this it?

So overall, it would be:

2vΔv = 2uΔu + 2as(Δa/a + Δs/s)

Is this correct?

• Delta2
haruspex
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So overall, it would be:

2vΔv = 2uΔu + 2as(Δa/a + Δs/s)

Is this correct?
Yes, but you can simplify that with some cancellation.

Thank you, I am very clear about it now.
The end result will be:
2vΔv=2uΔu + 2sΔa + 2aΔs

Thanks all for the help.

• Delta2