First slide

Accuracy; precision of instruments and errors in measurements

Question

A body travels uniformly a distance of $(S + Δ S)$ in a time $(t \pm Δ t)$ . what may be the condition so that body within the error limits move with a velocity $(\frac{S}{t} \pm \frac{Δ S}{Δ t}) ?$

Moderate

A

$\frac{Δ t}{t} + \frac{S (Δ t)^{2}}{(Δ S) t^{2}} = \pm 1$
B

$\frac{Δ t}{t} + \frac{S Δ t}{Δ S t} = \pm 1$
C

$\frac{Δ t}{t} + \frac{(Δ S) t}{S (Δ t)} = \pm 1$
D

$\frac{Δ t}{t} + \frac{S^{2} Δ t}{(Δ S)^{2} t} = \pm 1$

Solution

$\begin{array}{l} V = \frac{S}{t} \\ Δ V = \frac{1}{t} \cdot \frac{\partial S}{\partial S} \cdot Δ S + \frac{S}{t^{2}} Δ t = (\frac{Δ S}{t} + \frac{S Δ t}{t^{2}}) \end{array}$
$\begin{array}{l} So \frac{Δ S}{t} + \frac{S Δ t}{t^{2}} = \frac{Δ S}{Δ t} \cdot \frac{Δ t}{t} + \frac{(Δ S) \cdot S (Δ t)^{2}}{(Δ S) \cdot t^{2} (Δ t)} \\ or \frac{Δ S}{Δ t} [\frac{Δ t}{t} + \frac{S (Δ t)^{2}}{(Δ S) t^{2}}] = \pm \frac{Δ S}{Δ t} (given) \\ So, \frac{Δ t}{t} + \frac{S (Δ t)^{2}}{(Δ S) t^{2}} = \pm 1 \end{array}$

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