Question

In the following figure, a body starts from G, and reaches D via A, B and C.


Calculate the ratio of the total distance travelled to the total displacement of the body if the curves GA and BC have the same radius, and all the lengths given are in feet.

• A. $\frac{35}{53}$
• B. $\frac{11}{53}$
• C. $\frac{66}{35}$
• D. $\frac{53}{35}$

Answer 1

The correct option is D $\frac{53}{35}$

Given figure:

The distance travelled from G to D, $d=\text{GA}+\text{AB}+\text{BC}+\text{CD}$
The curves $\text{GA}$ and $\text{BC}$ are quarter sections of a circle of radius $50\text{ft}$.
Their length = $\frac{1}{4}\times 2\pi \times 50$
$⟹\text{GA}=\text{BC}=25\pi \text{ft}$
Also, $\text{AB}=\text{DF}-50-50=400-100$
$⟹\text{AB}=300\text{ft}$
$\text{CD}=\text{GF}=300\text{ft}$
$\therefore d=(2\times 25\pi )+(2\times 300)$
$d=(50\pi +600)\text{ft}$
$=50\pi +600=50\left(\frac{22}{7}\right)+600$
$=\frac{5300}{7}\text{ft}......\left(i\right)$
The displacement from G to D is $\text{GD}$.
Using Pythagoras theorem, $\text{GD}=\sqrt{(\text{GF}{)}^2+(\text{FD}{)}^2}$
$⟹\text{GD}=\sqrt{(300{)}^2+(400{)}^2}$
$⟹\text{GD}=500\text{ft}......\left(ii\right)$

$\therefore \frac{distance}{displacement}=\frac{\frac{5300}{7}}{500}=\frac{5300}{3500}=\frac{53}{35}$