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?

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

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.

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.

(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 $1.2\times {10}^{2}m/s^2$.

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 $12g$.

(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$1.4\times {10}^{8}N$

(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.