"This is the beginning of a long journey for you and me."
"For our people to be the strongest in our Galaxy, we had to weed out our weak ones."
"No one is going to pull their punches. This isn't like the cartoons you watch on Saturday morning. The p...
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"A gun is designed that can launch a projectile of mass 10 kg at a speed of 200 m s−1 . The gun is placed close to a straight, horizontal railway line and aligned such that the projectile will land further down the line. A small rail car of mass 200 kg and travelling at a speed of 100 m s−1 passes the gun just as it is fired. Assuming the gun and the car are at the same level, at what angle upwards must the projectile be fired so that it lands in the rail car?"
The second the question left Liz Allan's lips, Mira slammed her hand onto the red button in front of her, quickly managing to beat Peter who had reached for the buzzer too at the same time.
Maybe she was using her powers a little during tryouts for the Decathlon team, but she was too focused on the answer ringing through her ears to care.
The second Liz handed her the chalk with a bright smile, Mira made her way toward the board "To answer this, we need to know the distance around Pluto's equator and the crawling speed of a mauve caterpillar. Since we are provided with the radius of Pluto in the question, we can calculate the distance via the formula 2πr."
Her hands were moving against the board quickly and with ease, drawing physics equation after equation.
"The crawling speed of a mauve caterpillar, however, is considerably more difficult to find. The first thing to do is to replace as many words with symbols as possible. We could use 'm' to represent the length of a mauve caterpillar, 'v' to represent the length of a violet caterpillar, and 'l' to represent the length of a lavender caterpillar, but then what symbol would we use to represent the crawling speed of each? It will be easier to use standard symbols for quantities and use subscripts to indicate which caterpillar we are writing about. Thus, the length of a mauve caterpillar will be lm and the crawling speed of a lavender caterpillar will be vl . Our five pieces of information then become-"
She drew a few symbols on the board. (a) 5lm = 7lv
(b) 3ll + lm = 8lv
(c) 5ll + 5lm + 2lv = 1
(d) vl = lv/10
(e) vv = 2vl and vm = 2vl .
"Where we are told something is "as long as" something else, that is the same as saying they are equal. "Twice" means a factor of two. The fourth fact, which - unlike the other facts - contained information about time taken, has now become about speed and distance (using the fact that speed is distance divided by time taken), to make it more similar to the other four points. Given that we are trying to find the speed of a mauve caterpillar, it makes sense to start with the second half of the fifth fact, vm = 2vl , and combine it with the fourth fact: vm = 2vl (1.12) vm = 2 lv 10 (1.13) So in order to find the speed of a mauve caterpillar vm, we need to find lv, the length of a violet caterpillar. "
Mira momentarily glanced over her shoulder at the other students in the room, the older kids just staring at her in shock, Liz grinning with excitement and Peter smiling back at her proudly. Even Michelle in the corner of the room was fighting a smile as she watched Mr Harrison completely sputter unable to comprehend what she was saying right now. Why he was the teacher running the Decathlon team no one had any idea.
"6 Facts (a)-(c) all contain a term in lv. Looking at these facts, they represent three equations with three unknowns (lm, lv and ll) and so are a set of simultaneous equations we can solve for lv. Rearranging fact (a) gives that lm = 7lv/5, and substituting this into fact (b) gives: 3ll + 7lv 5 = 8lv (1.14) which can be solved for an expression for ll in terms of lv: ll = 33 15 lv (1.15) You should not be put off if, in your solution, you encounter some more unusual fractions than you might expect in other exams. Substituting our expressions for lm and ll into fact (c) then gives 5 33 15 lv + 5 7 5 lv + 2lv = 1 (1.16) Simplifying: 11lv + 7lv + 2lv = 20lv = 1 (1.17) and we can rearrange this for lv to find that lv = 1 20 = 0.05 m (1.18) Since, by equation 1.13, vm = 2lv/10, this means that the speed of the mauve caterpillar vm is vm = 2 × 0.05 10 = 0.01 m s−1 (1.19) Now that we have vm we can determine the final answer. The circumference of Pluto is C = 2πr, where r is given to us in the question as r = 1180 km. If we didn't have a calculator, estimating π ≈ 3 gives: C ≈ 2 × 3 × 1200 = 7200 km (1.20) Note that we've approximated 1180 km as 1200 km, but we also "underestimated" the value of π so we've got a reasonable answer. The time take to crawl is then given by t = C vm = 7200 × 103 1 × 10−2 = 7.2 × 108 seconds ≈ 22.8 years (1.21) If we were using a calculator we'd get t = 7.4 × 108 s ≈ 23.5 years, which is less than 3% different."
