The diagram shows the orbit of a comet around the Sun. The comet takes 84 years to complete one full orbit. Calculate the time period of the comet’s orbit in seconds. Use a value of 365 for the number of days in a year. Give your answer in scientific notation to two decimal places. At which point in the comet’s orbit will its speed be the greatest? (A) When it is farthest from the Sun. (B) When it is closest to the Sun.
Let’s begin with the first part of the question and write the comet’s period in seconds. Recall that an object’s orbital period is the amount of time it takes to complete one full orbit. Here, we know it takes 84 years for this comet to complete one full orbit. So this is its orbital period. We want to calculate the comet’s period in seconds. So let’s do some converting.
On the way from years to seconds, we’ll recall some equivalences in more intermediate time units such as days, hours, and minutes. We’ll start with the conversion from years to days. And remember that there are 365 days in one year. We can use this equivalence, one year equals 365 days, to write this conversion factor 365 days divided by one year. Because the numerator and denominator have the same value, this whole term, this conversion factor, is just equal to one. So when we multiply it by our orbital period, we can introduce new units without changing its value at all. Thus, we have 84 years times 365 days divided by one year.
Because units of years appear in the original term and in the denominator of the conversion factor, they get canceled out, leaving only units of days associated with our value. But before we grab our calculators and start multiplying, let’s set up the rest of the conversion factors to get from days to seconds. The next stop in our journey is days to hours. So let’s recall that one day equals 24 hours. And here is our next conversion factor.
It’s worth noting that days should be in the denominator here because we’re trying to cancel units of days from the numerator of the previous conversion factor. The units do cancel. And if we were to multiply 84 by 365 by 24 hours, we would have the orbital period of the comet written in hours. But we’re not done yet.
Next, we’ll convert from hours to minutes. There are 60 minutes in an hour. So here’s our next conversion factor. Having hours in the denominator cancels out units of hours from the whole expression, leaving only units of minutes. We have one more conversion factor to write until we’ve reached units of seconds. There are 60 seconds in a minute. And writing our final conversion factor, we’re able to cancel out units of minutes, leaving seconds as the only unit associated with this entire quantity.
Finally, grabbing our calculators and clearing some room on screen, we have 84 times 365 times 24 times 60 times 60 seconds, which gives us 2,649,024,000 seconds. A number this big is way better off written in scientific notation. So let’s move the decimal place one, two, three, four, five, six, seven, eight, nine places to the left. Now, we have 2.649024 times 10 to the nine seconds. And rounding this figure to two decimal places, we found that the orbital period of the comet is 2.65 times 10 to the nine seconds.
Now, let’s move on to the second part of the question. At which point in the comet’s orbit will its speed be the greatest? Unlike for a circular orbit, when an object has an elliptical orbit, like the comet does, its speed does not remain constant. Rather it varies depending on where the object is in its orbit. Recall that gravity is the force that keeps an object in orbit and that the gravitational force is greater when the objects are closer together than when they’re farther apart.
The entire time that the comet is approaching the Sun over here, the gravitational force on the comet is getting stronger and stronger, causing it to accelerate and become faster and faster. Thus, by the time the comet is passing right by the Sun, it’s been speeding up for years and years and reaches its maximum speed. The comet has a great initial speed when it starts moving away from the Sun. But on this entire leg of its journey, gravity is acting on the comet, pulling it back towards the Sun and causing it to slow down. Eventually, the comet slows down so much that it’s overcome by the Sun’s gravitational pull. So it starts to approach the Sun, picking up speed along the way and the cycle continues.
This agrees with answer choice (B). The comet is in elliptical orbit around the Sun. And therefore, its speed will be the greatest when it is closest to the Sun.