The energy of a system as a function of time is given as where . The measurement of has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of at is _____. [2015]

Question

The energy of a system as a function of time $t$ is given as $E (t) = A^{2} \exp (- α t),$ where $α = 0.2 s^{- 1}$ . The measurement of $A$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $E (t)$ at $t = 5 s$ is _____. [2015]

Smart Achievers · Accepted Answer

(4)

$E_{(t)} = A^{2} e^{- \propto t} = A^{2} e^{- 0.2 t}$

$∴$ $\log_{e} E = 2 \log_{e} A - 0.2 t$

On differentiating we get $= \frac{d E}{E} = 2 \frac{d A}{A} - 0.2 \frac{d t}{t} \times t$

As errors always add up

$∴$ $\frac{d E}{E} \times 100 = 2 (\frac{d A}{A} \times 100) + 0.2 t (\frac{d t}{t} \times 100)$

$\Rightarrow \frac{d E}{E} \times 100 = 2 \times 1.25 % + 0.2 \times 5 \times 1.5 % = 4 %$

Solution

Q.
The energy of a system as a function of time $t$ is given as $E (t) = A^{2} \exp (- α t),$ where $α = 0.2 s^{- 1}$ . The measurement of $A$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $E (t)$ at $t = 5 s$ is _____. [2015]

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Solution

Q. The energy of a system as a function of time t is given as E(t)=A2exp(-αt), where α=0.2 s-1. The measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of E(t) at t=5 s is _____. [2015]

Ans. (4) E(t)=A2e-∝t=A2e-0.2t ∴ logeE=2logeA-0.2t On differentiating we get =dEE=2dAA-0.2dtt×t As errors always add up ∴ dEE×100=2(dAA×100)+0.2t(dtt×100) ⇒ dEE×100=2×1.25%+0.2×5×1.5%=4%

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Q.
The energy of a system as a function of time $t$ is given as $E (t) = A^{2} \exp (- α t),$ where $α = 0.2 s^{- 1}$ . The measurement of $A$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $E (t)$ at $t = 5 s$ is _____. [2015]