Question

A body travels uniformly a distance of (13.8 + 0.2) meter in a time (4.0 + 0.3) second. Find the velocity of the body within error limits and the percentage error?

Answer 1

Hint: The result of every measurement by any measuring method contains some uncertainty, which is called an error. In order to calculate the percentage error first calculate the error limits in velocity by the formula $\dfrac{{\Delta v}}{v} = \dfrac{{\Delta d}}{d} + \dfrac{{\Delta t}}{t}$ then multiply the calculated value by hundred.
Formula used:
$velocity = \dfrac{\text{Distance}}{\text{Time}} \Rightarrow v = \dfrac{d}{t}$, $\dfrac{{\Delta v}}{v} = \dfrac{{\Delta d}}{d} + \dfrac{{\Delta t}}{t}$

Complete step-by-step solution:
Error limits in velocity is $\dfrac{{\Delta v}}{v} = \dfrac{{\Delta d}}{d} + \dfrac{{\Delta t}}{t}$ where $\Delta d$ is the distance error and d is the total distance and $\Delta t$ is the time error and t is the total time.
Given that,
Distance = (13.8 + 0.2) meter
Time = (4.0 + 0.3) second
Distance error $\Delta d$ = 0.2 meter
Time error $\Delta t$ = 0.3 second
Therefore velocity=$\dfrac{\text{Distance}}{\text{Time}}=\dfrac{{13.8}}{4} =3.45\dfrac{m}{s}$
Error limits in velocity = $\dfrac{{\Delta v}}{v} = \dfrac{{\Delta d}}{d} + \dfrac{{\Delta t}}{t}$
= $\dfrac{{0.2}}{{13.8}} + \dfrac{{0.3}}{{4.0}}$
= $0.089$
$\Rightarrow \dfrac{{\Delta v}}{v} = $ 0.089
Therefore change is velocity = $\Delta v = 0.089 \times 3.45 = 0.30$
Hence Velocity within error limits = $(3.45 \pm 0.3)\dfrac{m}{s}$
Percentage error = $\dfrac{{\Delta v}}{v} \times 100 = 0.089 \times 100 = 8.9\% $

Note: In this question first we calculated the velocity by dividing the distance by the time it takes to travel that same distance after that we calculated the error limits in velocity, hence the velocity within error limits is calculated to be as $(3.45 \pm 0.3)\dfrac{m}{s}$ with a percentage error of $8.9\%.$