Question

An interstellar ship has a mass of $\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{k}\mathit{g}$and is initially at rest relative to a star system. (a)What constant acceleration is needed to bring the ship up to a speed of 0.10c(where cis the speed of light, $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$) relative to the star system in 3.0days? (b) What is that acceleration in gunits? (c) What force is required for the acceleration? (d) If the engines are shut down when 0.10 cis reached (the speed then remains constant), how long does the ship take (start to finish) to journey 5.0light-months, the distance that light travels in 5.0months?

Short answer

1. Acceleration needed to bring ship to speed of 0.10 c is$1.2x{10}^{2}m/s^2$
2. Acceleration in g unit is 12g
3. Force required for acceleration $1.4x{10}^{8}N$
4. Time taken by ship to travel 5.0 light months with speed 0.10 c is 4.2 years

Step-by-step solution

Given

1. $m=1.20x{10}^6\mathrm{kg}$,
2. V=0.10 c,
3. $c=3.0x{10}^8\mathrm{m}/\mathrm{s}$
   

Understanding the concept

Use the formula for the acceleration to find acceleration. For force we use Newton’s second law.

Find time taken by ship to travel 5.0light month using velocity equation.

Formulae:

1. $\mathit{a}\mathbf{=}\frac{\mathbf{∆}\mathbf{V}}{\mathbf{∆}\mathbf{t}}$
   
2. $\mathbf{F}\mathbf{=}\mathbf{ma}$
   
3. role="math" localid="1660901234819" $\mathit{V}\mathbf{=}\frac{\mathbf{d}}{\mathbf{t}}$
   

Here, F is force, a is acceleration of the object, m is mass of the object,$\mathbf{∆}\mathbf{V}$ is change in velocity,$\mathbf{∆}\mathit{t}$ is change in time, d is displacement, t is time.

Calculate the time in seconds

Converting time from day to seconds,

$$\begin{gathered} ∆\mathrm{t}=3\mathrm{days}\times \frac{24\mathrm{h}}{1\mathrm{day}}\times \frac{3600\mathrm{s}}{1\mathrm{h}} \\ =259200\mathrm{s} \end{gathered}$$

Therefore, the change in velocity takes place in time 259200 s.

 Calculate the acceleration needed to bring ship to speed of 0.10 c

(a)

The acceleration is the change in velocity with respect to time. Therefore,

$$\begin{gathered} a=\frac{∆V}{∆t} \\ =\frac{\left(0.10\right)\left(3\times {10}^{8}m/s\right)}{259200s} \\ =1.2\times {10}^{2}m/s^2 \end{gathered}$$

Therefore, the acceleration is $1.2\times {10}^{2}m/s^2$.

 Calculate the acceleration in g units

b)

To calculate the acceleration in g units, divide the acceleration by g and multiply it by g.

$$\begin{gathered} a=\left(\frac{a}{g}\right)g \\ =\frac{1.2\times {10}^{2}m/s^2}{9.8m/s^2}g \\ =12g \end{gathered}$$

Therefore, the acceleration in the units of g is $12g$.

 Calculate the force required for the acceleration

(c)

Use Newton’s law of motion to calculate the force required for the acceleration.

$\begin{array}{lll}F=ma& & \\ =(1.20\times {10}^6& \mathrm{kg}\left)\right(1.2\times {10}^2& m/s^2)\\ =1.4\times {10}^{8}N& & \end{array}$

Therefore, the force is$1.4\times {10}^{8}N$

 Calculate the time required to travel 5.0 light months

(d)

Use the formula for the velocity to calculate the time required to travel 5.0 light months.

$$\begin{gathered} t=\frac{d}{V} \\ =\frac{5.0cmonths}{0.1c} \\ =50months \\ \approx 4.2years \end{gathered}$$

The time required to travel 5.0 light months is 4.2 years.