Question

What is the order of magnitude of the distance of a quasar from us if light takes $2.9$billion years to reach us?
A) $2.7 \times {10^{25}}m$
B) ${10^{25}}m$
C) ${10^{24}}m$
D) $3 \times {10^{25}}m$

Answer 1

Hint:Recall that velocity is defined as the rate of change of position or displacement of any body or object with time. It is a vector quantity. It has both magnitude and direction. In order to change the velocity of an object, its position should be changed. The velocity of an object changes with change in the direction of the object.

Complete step by step answer:
Step I:
Given that the time taken by light to reach is $ = 2.9$billion years
Or $2.9 \times {10^9}$years
Converting years into seconds,
$t = 2.9 \times {10^9} \times (365 \times 24 \times 60 \times 60)s$
$\Rightarrow t = 9.14 \times {10^{16}}s$
Step II:
The formula for velocity is written as
$Velocity = \dfrac{{Displacement}}{{Time}}$
Since only the magnitude of the distance travelled by light is to be calculated, therefore considering the distance only, as it is a scalar quantity.
Or $Dis\tan ce = Velocity \times Time$---(i)
Step III:
The velocity with which the light travels is $3 \times {10^8}$
Substitute the values of velocity and time in equation (i), and solve
$d = 3 \times {10^8} \times 9.14 \times {10^{16}}$
$\Rightarrow d = 27.42 \times {10^{24}}$
$\Rightarrow d = 2.74 \times {10^{25}}$
Step IV:
Since $2.74 \times {10^{25}}$ is less than $5$, therefore it can be taken as $1$.
Therefore the order of the magnitude of distance will be $1 \times {10^{25}} = {10^{25}}m$

Hence, Option B is the right answer.

Note:It is important to note that the term displacement and distance are different. Distance is the total length between any two points and it does not consider direction. It can only have magnitude with positive values and can be measured along curved or straight paths. Displacement is the shortest path measured between any two points. It depends on both magnitude and direction. It’s magnitude can be positive, zero or negative and can be measured only in straight paths.