Solved Problem on One-dimensional Motion

FISICAEXE

The biggest known star (until June 2019) is VY Canis Majoris in the constellation of Canis Major, with an estimated diameter of 1,975,000,000 km. Making the (absurd) assumption that a commercial jet could fly close to the surface of the star at a constant speed of 990 km / h. How long would the jet take for a spin on the star. Answer in years.

Problem data:

star Diameter: D = 1,975,000,000 km;
jet Speed: v = 990 km/h.

Problem diagram:

Figure 1

Solution:

The distance that the airplane should travel is the circumference of the star. The length of a circle is given by

\[ \begin{gather} C=2\pi r \end{gather} \]

where, r is the radius of the circumference, the diameter D of a circle being equal to 2r, the length of a circle can also be calculated by

\[ \begin{gather} C=\pi D \end{gather} \]

using, π = 3.14 the circumference of the star will be

\[ \begin{gather} C=3.14 \times 1975000000\;\mathrm{km} \\[5pt] C=6201500000\;\mathrm{km} \end{gather} \]

As the plane speed is constant, we have that travel time is given by

\[ \begin{gather} \bbox[background:#99CCFF; padding:10px; border-radius:10px;] {v=\frac{\Delta S}{\Delta t}} \end{gather} \]

\[ \begin{gather} \Delta t=\frac{\Delta S}{v} \end{gather} \]

using the speed value given in the problem and the distance being the length of the circumference calculated above

\[ \begin{gather} \Delta t=\frac{6201500000\;\mathrm{\cancel{km}}}{990\;\mathrm{\frac{\cancel{km}}{h}}} \\[5pt] \Delta t=6264141\;\mathrm{h} \end{gather} \]

Converting this value to years as it asks for the problem

\[ \begin{gather} \Delta t=6264141\;\cancel{\mathrm{hours}}\times\frac{1\;\cancel{\mathrm{day}}}{24\;\cancel{\mathrm{hours}}}\times\frac{1\;\mathrm{year}}{365\;\cancel{\mathrm{days}}} \end{gather} \]

\[ \begin{gather} \bbox[background:#FFCCCC; padding:10px; border-radius:10px;] {\Delta t=715\;\mathrm{years}} \end{gather} \]

Note: for comparison, the Sun is 1,391,000 km in diameter, under the same conditions, the airplane would take "only" 184 days to go around the Sun.