Chapter 24, Problem 16Q

(a)

To determine

The time taken by the blob to move from A to B if the distance from point A to point B in the following figure is $26\text{ly}$, and the blob moves with a speed, $\frac{13}{15}$ times the speed of light. During the motion of the blob from A to B, it moves $24\text{ly}$ toward the Earth and $10\text{ly}$ in a side ways, transverse direction.

Answer to Problem 16Q

Solution:

$30\text{ years}$.

Explanation of Solution

Given data:

Distance between point A and point B is $26\text{ly}$.

Speed of the blob is $\frac{13}{15}c$. Here, c is speed of light.

Blob moves toward Earth from point A through a distance $24\text{ly}$.

Blob moves in a side ways, transverse direction through a distance $10\text{ly}$.

Formula used:

The expression for the speed of any object is written as,

$v=\frac{d}{t}$

Here, $v$ is the speed of the object, $d$ is the distance and $t$ is the time taken to travel that distance.

Explanation:

Write the expression for the speed of blob.

$v=\frac{d}{t}$

Rearrange this for $t$.

$t=\frac{d}{v}$

Substitute $26\text{ly}$ for $d$ and $\frac{13}{15}c$ for $v$.

$\begin{array}{c}t=\frac{\left(26\text{ly}\times \raisebox{1ex}{$9.5\times {10}^{15}\text{m}$}\!\left/ \!\raisebox{-1ex}{$1\text{ly}$}\right.\right)}{\left(\frac{13}{15}\times 3\times {10}^8\text{m/s}\right)}\\ =\left[\left(95\times {10}^7\text{s}\right)\times \frac{1\text{year}}{60\times 60\times 24\times 365\text{s}}\right]\\ =30\text{ years}\end{array}$

Conclusion:

Therefore, the time taken by blob to move from A to B is $30\text{ years}$.

(b)

To determine

The year in which the light from B reaches Earth if the distance from A to point B in the following figure is $26\text{ly}$ and that the blob moves with a speed $\frac{13}{15}$ times the speed of light. During the motion of blob from A to B, it moves $24\text{ly}$ toward Earth and $10\text{ly}$ in a side ways, transverse direction, whereas the light from the blob at A reaches Earth in $2020$.

Answer to Problem 16Q

Solution:

$2026$.

Explanation of Solution

Introduction:

If any object appears to be moving faster than the speed of light, the motion of that object is called superluminal motion. Some astronomical objects such as quasars, blazars, radio-galaxies show superluminal motion.

Explanation:

It is given that the light from the blob at A reaches Earth in $2020$. The distance of the Earth from point B is $24\text{ly}$ less than the distance of the Earth from point A. The first pulse of light from point A reaches Earth in $2020$. The second pulse of light is emitted from point B after $t=30\text{y}$. The second pulse has to travel $24\text{ly}$ less distance to reach the Earth. The second pulse reaches the Earth $6\text{ years}$ after the first pulse reaches. So, the second pulse reaches the Earth in $2026$.

Conclusion:

Therefore, the year in which the light from B reaches the Earth is $2026$.

(c)

To determine

The apparent speed of the the blob moving across the sky as seen from the Earth if the distance from point A to point B in the following figure is $26\text{ly}$ and the blob moves with a speed, $\frac{13}{15}$ times the speed of light. During the motion of blob from A to B, it moves $24\text{ly}$ toward the Earth and $10\text{ly}$ in a side ways, transverse direction.

Answer to Problem 16Q

Solution:

The speed of the the blob moving across the sky appears to be 1.7 times the speed of light.

Explanation of Solution

Given data:

Distance between two points in the sky is $10\text{ly}$.

Time taken by light to travel from one point to another is $6\text{y}$.

Formula used:

The expression for the speed of any object is written as,

$v=\frac{d}{t}$

Here, $v$ is the speed of the object, $d$ is the distance and $t$ is the time taken to travel the distance.

Explanation:

Write the expression for the speed of blob.

$v=\frac{d}{t}$

Substitute $10\text{ly}$ for $d$ and $6\text{y}$ for $t$.

$\begin{array}{l}v=\frac{10\text{ly}\times \frac{c\text{y}}{\text{1}\text{ly}}}{6\text{y}}\\ v=1.7c\end{array}$

Conclusion:

Therefore, the apparent speed of the the blob is 1.7 times the speed of light.