Question

The distance $S$ traveled by a particle is calculated using the formula $S=ut-\frac{1}{2}a{t}^2$. If there is $1\%$ error in $t$, find the approximate percentage error in $S$.

Answer 1

$s=ut-\frac{1}{2}a{t}^2$ $\to \left(1\right)$

$s=u(t\pm \frac{1}{100})-\frac{1}{2}a{(t\pm \frac{1}{100})}^2$

$=ut\pm \frac{u}{100}-\frac{1}{2}a(t^2\pm \frac{2t}{100}+\frac{1}{10000})$

$s=ut\pm \frac{u}{100}-\frac{1}{2}a{t}^2\pm \frac{at}{100}+\frac{a}{20000}$ ………..$\left(2\right)$

Now if we take $a=10$, $u=50$, $t=100$

$s=50\times 100-\frac{1}{2}\times 10\times 100\times 100=5000-50000$

$=45000$

Similarly

$s=50\times 100\pm \frac{50}{100}-\frac{1}{2}\times 10\times 100\times 100\pm \frac{10\times 100}{100}+\frac{100}{20000}$

$=500\pm 0.5-50000\pm 10+\frac{1}{200}$

$=500.5-50010+\frac{1}{200}$

$=500.5-50010.005$

$=-49,509.505$

$\therefore 45000$ $100\%$

$49509$ x

$\therefore \frac{49509\times 100}{45000}$

$={110}^o$.

$\therefore$ Change in S is $10\%$.