Chapter 24, Problem 16Q

To determine

(a)

The time taken by the blob to travel from the point $A$ to $B$.

Answer to Problem 16Q

The time taken by the blob to travel from point $A$ to $B$ is $30\text{}\text{years}$.

Explanation of Solution

Given:

The distance between the point $A$ and $B$ is, $d_1=26\text{}\text{ly}$.

The distance travelled by the blob in sideway direction is, $d_2=24\text{ly}$.

The distance travelled by the blob in the transverse direction is, $d_3=10\text{ly}$.

Formula used:

The expression for the calculation of the time is given by,

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

Here, $t$ is the time taken, $d$ is the distance covered and $v$ is the speed.

Calculation:

The speed of the blob is calculated as,

$\begin{array}{c}v=\left(\frac{13}{15}\right)\left(3\times {10}^8\text{m}/\text{s}\right)\\ =\left(\frac{13}{15}\right)\left(3\times {10}^8\text{m}/\text{s}\times \frac{{10}^{-3}\text{km}}{1\text{m}}\right)\\ =\left(\frac{13}{15}\right)\left(3\times {10}^5\text{km}/\text{s}\right)\\ =\left(\frac{13}{15}\right)\left(3\times {10}^5\text{km}/\text{s}\times \frac{\left(60\times 60\times 24\times 365\right)\text{s}}{1\text{year}}\right)\end{array}$

Solve further,

$\begin{array}{c}v=\left(\frac{13}{15}\right)\left(9.46\times {10}^{12}\text{km}/\text{year}\right)\\ =8.19\times {10}^{12}\text{km}/\text{year}\end{array}$

The time taken by the blob the point from point $A$ to $B$ is calculated as,

$\begin{array}{c}t=\frac{d_1}{v}\\ =\frac{26\text{ly}}{8.19\times {10}^{12}\text{km}/\text{year}}\\ =\frac{\left(26\text{ly}\times \frac{9.46\times {10}^{12}\text{km}}{1\text{ly}}\right)}{\left(8.19\times {10}^{12}\text{km}/\text{year}\right)}\\ =30\text{years}\end{array}$

Conclusion:

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

To determine

(b)

The year in which the light from the point $B$ reaches the earth.

Answer to Problem 16Q

The year in which the light from the point $B$ reaches the earth is $2023$.

Explanation of Solution

Given:

The year in which the light from the point $A$ reaches the earth is, $2020$.

The distance between the point $A$ and $B$ is, $d_1=26\text{}\text{ly}$.

The distance travelled by the blob in sideway direction is, $d_2=24\text{ly}$.

The distance travelled by the blob in the transverse direction is, $d_3=10\text{ly}$.

Formula used:

The distance covered by the light in the sideway direction is calculated as,

$t=\frac{d_2}{v}$

The extra time taken by the light to travel from point $B$ is given by,

$\begin{array}{c}\text{extra time}=\left(\text{time taken from point }A\right)-\left(\text{time taken from point }B\right)\\ =30\text{years}-27\text{years}\\ =\text{3 years}\end{array}$

The year in which the light from the point $B$ reaches the earth is given by,

$\text{year of light reaching from point }B=\left(\begin{array}{l}\text{year of light reaching the }\\ \text{earth from the point }A\end{array}\right)+\left(\begin{array}{l}\text{extra}\\ \text{time}\end{array}\right)$

Calculation:

The distance covered by the light in the sideway direction is calculated as,

$\begin{array}{c}t=\frac{d_2}{v}\\ =\frac{24\text{ly}}{8.19\times {10}^{12}\text{km}/\text{year}}\\ =\frac{\left(24\text{ly}\times \frac{9.46\times {10}^{12}\text{km}}{1\text{ly}}\right)}{\left(8.19\times {10}^{12}\text{km}/\text{year}\right)}\\ =27\text{years}\end{array}$

The extra time taken by the light to travel from point $B$ is calculated as,

$\begin{array}{c}\text{extra time}=\left(\text{time taken from point }A\right)-\left(\text{time taken from point }B\right)\\ =30\text{years}-27\text{years}\\ =\text{3 years}\end{array}$

The year in which the light from the point $B$ reaches the earth is calculated as,

$\begin{array}{c}\text{year of light reaching from point }B=\left(\begin{array}{l}\text{year of light reaching the }\\ \text{earth from the point }A\end{array}\right)+\left(\begin{array}{l}extra\\ time\end{array}\right)\\ =2020+3\\ =2023\end{array}$

Conclusion:

Therefore, the year in which the light from the point $B$ reaches the earth is $2023$.

To determine

(c)

The speed of the blob across the sky on the earth.

Answer to Problem 16Q

The speed of the blob across the sky on the earth is $\frac{10}{3}c$.

Explanation of Solution

Given:

The year in which the light of the blob from the point $A$ reaches the earth is, $y_1=2020$.

The year in which the light of the blob from the point $B$ reaches the earth is, $y_1=2023$.

The distance travelled by the blob in the transverse direction is, $d_3=10\text{ly}$.

Formula used:

The expression for the calculation of the speed of the blob across the sky is given by,

$\text{speed of blob across the sky}=\frac{\left(\begin{array}{l}\text{transverse distance between }\\ \text{the point A and B}\end{array}\right)}{\left(\begin{array}{l}\text{difference in time taken by the}\\ \text{ light from A and B to reach earth}\end{array}\right)}$

Calculation:

The speed of the blob across the sky on the earth is calculated as,

$\begin{array}{c}\text{speed of blob across the sky}=\frac{\left(\begin{array}{l}\text{transverse distance between }\\ \text{the point A and B}\end{array}\right)}{\left(\begin{array}{l}\text{difference in time taken by the}\\ \text{ light from A and B to reach earth}\end{array}\right)}\\ =\frac{10\text{ly}}{\left(2023-2020\right)\text{years}}\\ =\frac{10}{3}c\end{array}$

Conclusion:

Therefore, the speed of the blob across the sky on the earth is $\frac{10}{3}c$.