Space Exploration: Choose Your Own Adventure!

Space Exploration: Choose Your Own Adventure!

It’s time to blast off! As the mathematics team of the space shuttle ______, do you have what it takes to boldly lead your crew where no one has gone before? Below is the first question of your journey. After solving each question, present your answer to the TCP staff at your table. If your answer is correct, you will be given the next “main” question. If it is incorrect, you may either skip the problem and be given a different question (a “side question”), or try again until you solve it correctly, after which you will move to the next main question. Scientific calculators are permitted. Bon voyage!

M-1

Penalty of 3 points to skip this problem.

Before we blast off, we need fuel! For this journey, we will be running on Oozle. Oozle is collected from exposure to hopes and dreams (HD). The maximum Oozle capacity of the space shuttle is 26.28 million units. One adult’s HD provides 6 units per hour, and one child’s HD provides 62 units \per hour. NASAL (New Westminster’s Awesome Space and Aeronautics controL) has three Oozle collectors:

1. (2 pts) Collector 1 is 65% efficient and is only compatible with the HD of elementary school children. It is placed at a location where there are fifteen school children for each adult office worker. If the HD of only the office workers at this location were harvested by a perfectly efficient collector, it would take about 25 years for it to fill the shuttle’s Oozle capacity alone. How many children are there? (Assume a year has 365 days.)
2. (2 pts) Collector 2 is 80% efficient and is compatible with both adults and children. However, from October 1st to March 31st, the HD of both adults and children is suppressed to half their usual capacity. Collector 2 harvests the HD of four adults and three children. How many units per hour are harvested from October to March?
3. (3 pts) Collector 3 is perfectly efficient as long as it harvests from no more than four people. If there are more people, its efficiency drops to 20%. It is placed in a house where there are five adults and five children, but four of the children are away at school from 8AM to 3PM and all the adults are away at work from 9AM to 5PM. The last child stays at home all day, but is accompanied by an adult babysitter at all times when it would otherwise be alone (but the babysitter is never in the house at the same time as the other adults or the rest of the children.) How many units are harvested per day, to the nearest tens place?
4. (1 pt) At 8AM on March 1st, 2017, all three collectors begin operating. What is the minimum number of days they must work before enough Oozle has been collected to fully fuel the shuttle? Round to the nearest whole number.

1. ______children b) ______units/hr c) ______units/day d) ______days
   

M-2

5pts. No penalty to skip this problem.

Now you’re ready to begin your journey. The first step is to leave the Solar System, and to do so you need to know the escape speed. Escape speed is the minimum speed your space shuttle needs to attain in order to escape the gravitational pull of the Sun.

Energy in a closed system is conserved. In other words, the sum of the rocket’s initial energy is equal to the sum of the rocket’s energy after it has escaped the gravitational pull.

Kinetic energy (K) is a type of energy that comes from movement.

K = ½mv2 where

• m is the mass of the moving object
• v is the object’s speed

To escape the pull of the Sun’s gravity, your shuttle must have a minimum of 0 joules of kinetic energy at an infinite distance from the Sun.

Potential energy (P) is a type of energy that comes from gravity.

P = -GMm/R where

• G is a constant equal to 6.67*10-11m3kg-1s-2
• M is the mass of the Sun
• R is the distance in metres between your rocket and the Sun.

As you can tell, at R = ∞, P = 0.

The Sun’s mass is 1.99*1030kg. The distance between the Earth and the Sun is 1.50*108km. In km/s, what minimum speed does the shuttle’s rocket need to escape the gravitational pull of the Sun? Assume that the shuttle does not have any other type of energy. Express your answer to the nearest whole.

______km/s

S2

2 points. Penalty of 2 points to skip this problem

Uh-oh...you fumbled it. Luckily, though you didn’t escape the Sun’s pull, your team managed to escape the Earth’s gravitational pull and enter your shuttle into orbit around the Sun (orbital velocity = 24 km/s). Now all you have to do is calculate the correct escape velocity from your place orbiting the Sun.

The force keeping you in orbit around the Sun is called the centripetal force, and it is calculated by mv2/R, where v is your shuttle’s velocity as it orbits and R is the distance between the shuttle and the center of the Sun. The centripetal force comes entirely from the pull of the Sun on your shuttle, meaning it is equal to the gravitational force, calculated by GMm/R2.

The Sun’s mass is 1.99*1030kg. The distance between the Earth and the Sun is 1.50*108km. G is a constant equal to 6.67*10-11m3kg-1s-2.What minimum speed do you need to add to your orbital velocity in order to escape the pull of the Sun? Express your answer to the nearest tenth of a km/s. HINT: Escape velocity is calculated by2GM/R.

______km/s

M3

No penalty to skip this problem.

On your way out of the Solar System, you detect a comet traveling in a straight line in the distance. Since you were a child, it’s been your dream to pit humankind’s technology against the awesome power of nature in a physics-defying situation! You prepare to chase down the comet. The speed of the comet is 15 km/s. The speed of the spaceship is 18 km/sec.Your respective positions are as shown, and you are both travelling in straight lines.

1. (2pts) In seconds, how long will you have to maintain your speed in order to catch up to the comet?

1. (4pts) As you consider the fulfillment of your greatest ambition, your hopes and dreams increase in magnitude at a constant rate, so that by the end of the race with the comet, your HD level is four times what it was at the start.. However, the navigator is unimpressed by this childish endeavour. His HD is inversely proportional to yours - that is, as your HD increases, his HD decreases in proportion so that the product of your HDs remains the same.

There is a perfectly efficient Oozle collector harvesting the HD of you and the navigator. At your regular HD levels (that is, the levels before the race began), you and the navigator would have been able to fill the Oozle collector 165 times in 100 days. At your HD levels at the end of the race, you would have been able to fill it 375 times in 100 days.

Afterwards, your navigator leaves in a huff. How long will it take for you to fill the empty collector alone, at your final HD level? Express your answer to the nearest hour.

1. ______seconds b) ______hours

S3

2pts; -2pts to skip.

Alas, the comet has bested you. But you’re not giving up! You’re going to prove the capacity of humankind by navigating through an asteroid belt! You encounter a comet moving 60km/s forward in the x direction, 90km/s backwards in the y direction, and 150km/s forward in the z direction. An asteroid is moving parallel to your ship at 20km/s backwards in the x direction and 30 km/s forward in the y direction. What is its total velocity with reference to your ship? Express to the nearest km/s.

M-4

6pts

No penalty to skip

After travelling for some time, you’re starting to run out of Oozle. Luckily, you’ve entered radio communication with a friendly alien spacecraft, whose crew has invited you to their planet Akennis to visit and refuel. However, your Universal Translator can only translate words, so you need to decipher their directions! The Akennisian number system is like ours except that they have a different base system and different symbols. Like ours, each self-contained symbol in their number system represents a different digit. For example, ] Д represents a number with two digits. The Akennisians use the same units of measurement as us and their base system is derived from the number of fingers they have.

Explanation of bases: We have ten fingers, and we use base ten. Base ten has ten digits, 0 through 9, and the number ten is represented with a 1 and a 0. If the Akennisians had a total of four fingers, they would use base 4. Base 4 has four digits, 1 through 3. The number four is represented as 10, and the number forty as 100.

Here are their messages to you:

• Hello! You must be new in the solar system. We’ve never seen another race with 」hands just like us before. Although we did notice you have ୭ fingers on each hand...we only have Ө on each hand, for a total of ] Д fingers.
• To get to our planet, first you need to face the right direction. Turn ⇬ Ө degrees to your left. You could also turn Ө ୭ Ө degrees to your right, of course. Same thing. There are ୭୭Д degrees in a circle, after all.

What direction must you face to go to Akennis? Express your answer in base 10, in degrees to the left or degrees to the right.

______degrees left / right

S4

2 points; -4 pts to skip

Uh-oh. Wrong direction. You return to your original starting location and inform the Akennisians that you couldn’t understand their directions. Somewhat perplexed, they agree to state the directions in a different way. Here is their new message to you. Disregard orbital mechanics; assume that traveling in space is akin to traveling in an air-resistance-free space near the surface of the Earth.

“Here’s a map of your current position. These are two circles with centers A and B that are tangent and congruent to each other. C is on the circumference of the circle with center B. Your spaceship is at B, facing A; Akennis is at C.”

Now, can you figure out the direction you need to take to reach the Akennisians’ planet, in degrees left or right?

______degrees left / right

M5

Each part worth 2 pts

No penalty to skip

You’re on your way to Akennis, the crew’s been working hard, and life is good. It’s time for a pizza and game break!

1. The vending machine sells custom made pizzas in the space lounge. To customize your pizza, you may choose two different toppings out of five available, and one base out of three available. How many choices do you have?
2. The crew decides to play a game with cards. 2 people draw a card from the deck, one by one, and the player with the more valuable card wins. (The king is most valuable, and the ace is least valuable. If the same value card is drawn, take suits into account: spade>heart>club>diamond). If Kevin draws a jack of hearts first, what is the probability that Kelsey will win against Kevin? Express your answer as a common fraction.
3. Kelsey and Kevin decide to play a game of Just Dance in the lounge. Kelsey has a 60% chance of winning a round against Kevin. If they play a best of 3 match (first to win 2 games wins), what is the probability that Kelsey will win? Express your answer as a percentage.

a)______choices b) ______c) ______%

M-6

No penalty to skip this problem.

It so happens that time passes more slowly for objects travelling at very high velocities than it does for stationary observers. This phenomenon is known as time dilation and has a formula:

t'=t *1-V2/c2 where:

• t’ is the amount of time that passes for the high-velocity object (this amount of time is dilated, or smaller),
• t is the amount of time that passes for the stationary observer
• V is the object’s velocity, and
• c is the speed of light.

1. (2pts) You’ve now been travelling for quite some time and your parents back on Earth are becoming concerned. They have sent a letter demanding that you contact them once a week.

Your shuttle travels at 80% the speed of light. From the reference frame of the shuttle, how often do you need to send updates for your parents to be satisfied? Express your answer in the format x days y hours. For example, if you needed to send updates every 1 day, 1 hour, and 31 minutes, you’d write 1 day 2 hours.

1. (4pts) When you were younger, your brother bragged, “I am 25% older than you”. Two years later, right before you left to explore the universe, he said, “I am now only 20% older than you”. It has been311/10 years since you left, and today you received a message from your brother, saying “If you had stayed at home, I would now be 13 ⅓ % older than you.” What has been your average speed in the 311/10 years that you have travelled? Express your answer as a decimal in terms of c.

1. ______b) ______

S-7

2 pts

Penalty of 2 points to skip this problem

Your mathematical errors have made your parents decide to come visit you. They get on a spaceship and fly in your direction. Starting from rest, they accelerate at a constant rate of 14 700 m/s2 until they reach your shuttle’s average speed, at which time they maintain that speed for another 5.2*105s. From the moment they leave, you fly your shuttle to meet them, also maintaining your average speed. After you travel 1.5876*1014m, your two shuttles meet. Now what is your shuttle’s average speed? Express your answer to two decimal places in terms of c. Note that c is 3*108s. Also note that for any quadratic ax2+bx+c=0, xis equal to (-bb2-4ac )/2a.

______

M8

Each part worth 3 pts.

No penalty to skip.

After travelling for some more time, you enter a new galaxy and decide to find a planet for maintenance and refuelling.

1. Your map says that of the forty million solar systems in the galaxy, 280 000 contain known hostile alien races, thirty million contain known friendly/neutral alien races, and the rest are unknown.

• Of the friendly/neutral races, thirty percent will never turn hostile under any circumstances, while the rest have a 0.3 percent chance of turning hostile.
• Of the hostile races, ninety percent will immediately take hostile action, five percent will take action after some time, and five percent will never take action (that is, remain peaceful).
• The unknown races have even chances of taking hostile action and remaining peaceful.

If the friendly/neutral, hostile, and unknown races are distributed evenly among the solar systems, what is the probability that the first race you encounter never takes hostile action? Express your answer as a percentage rounded to two decimal places.

1. You choose a solar system which contains 25 planets. Of these, 4 contain suitable repair shops, 14 are hostile, and 10 are physically capable of hosting human visitors. All planets with suitable repair shops are capable of hosting humans, and only one of these is hostile. 4 planets are non-hostile, but lack suitable repair shops and are incapable of hosting humans. If the number of hostile planets is equal to the number of planets that are either non-hostile or capable of sustaining humans, what is the probability that the first planet you pick is non-hostile, has a suitable repair shop, and can host human visitors? Assume that there is an equal probability of picking each planet, and express your answer as a percentage.

1. ______% b) ______%

S8

2pts; -2pts to skip

Oh, boy...you calculated wrong and picked a hostile planet. Luckily, they agree to drop hostilities if you help them fix their satellites...but first you’ve got to find them A system of Tracking and Data Relay Satellites is placed around their planet in geosynchronous orbit to help spacecraft communicate with their planet. Geosynchronous orbit means the satellites have the same orbital period as the planet, allowing them to remain at an almost fixed location over the planet. (A period is the time it takes, in seconds, to complete one repetition of a cycle, such as one complete orbit of a satellite or one complete rotation of the planet.)

The radius of the planet is 6371 km. The length of one day on the planet is 24 hours. Each satellite has an orbital speed that is 2.606 km/s faster than the speed of the planet’s rotation. What is the altitude (distance from the planet’s surface) of each TDRS? Express your answer to the nearest thousand kilometres.

______km

M9

8 pts

No penalty to skip

You’ve established good relations with the planet you’ve chosen, so you decide to dock and take your navigation equipment to get checked up. As it turns out, your idle curiosity about your average speed is going to serve you well. There’s been a mix-up at the repair shop - five crews from five different planets have had their virtual logbooks, radios, and payment receipts mixed up.

Each of the five sets of equipment involved come from a different planet:

Each of the virtual logbooks show different values for total distance travelled:

Each of the radios have different colours: