Intercepting a Meteor with a Rocket
A meteor is headed towards Earth so we need to intercept it.
A giant meteor is hurtling towards the centre of Brampton! The meteor must be intercepted so nobody is harmed. Luckily for us, The Canadian Space Agency is sending a rocket to destroy it, and with the use of simple grade 9 math, they can calculate and manipulate the interception point!
Meddling with the Rocket's Speed
The chart below shows the speeds (km per minute) and distances (km from the Earth's sea level) of the rocket the C.S.A. launched into space.
The Equation of the Line
y = the Distance of the Rocket from the Earth's Sea Level
x = the Speed of the Rocket
c (in this case 0, 20) = The Coordinates of the Speed and Distance at which the Rocket was Launched
m (in this case 0.5) = The Rate of Change/the Slope of the Line
The equation of a line is in the form y = mx + b. The equation of this line specifically is y = 0.5x + 20. However, the equation of a line can also be portrayed in standard form, and there are two standard forms. Ax + By = C, and Ax + By + C= 0. The equation of this line in the standard form Ax + By = C, is x - 2y = -40. The equation of this line in the other standard form Ax + By + C = 0, is x - 2y + 40 = 0. However, for both standard forms, A must be positive, and all values must be integers (no fractions).
There are multiple ways to figure out the equation of a line just by looking at this chart. One is to find the first difference of the dependent variable, which is the slope (y = mx + b). Then you find the dependent variable value when the independent variable is 0, and that is the y-intercept (y = mx + b). You can also find the equation of this line by subtracting any two dependent values by each other, and their corresponding independent values by each other, in the same order (i.e. 1020-20 and 2000-0). After this you divide the dependent variable by the independent variable, which gives you the slope (y = mx + b). And then you substitute any two corresponding coordinates into the equation. For example, 2020 = 0.5(4000) + b. This will give you 2020 = 2000 + b, and you simply subtract 2000 from 2020 afterwards, and you get b. To put the equation of the line into standard from, simply arrange y = mx + b into Ax + By = C, however A must be positive, and a whole number.
However, the C.S.A. did not need to find the equation of this line using any of these methods as they carefully planned and created the line before launching the rocket.
The chart below shows the speeds (km per minute) and distances (km from the Earth's sea level) of the meteor hurtling towards Earth.
The Equation of the Line
y = the Distance of the Meteor from the Earth's Sea Level
x = the Speed of the Meteor
b (in this case 0, 50,000) = The Coordinates of the Speed and Distance at which the Meteor were spotted
m (in this case -2) = The Rate of Change/The Slope of the Line
The equation of this line is y = -2x + 50,000. The equation of this line in standard form (Ax + By = C) is 2x + y = 50,000 and the equation of this line in standard form (Ax + By + C = 0) is 2x + y - 50,000 = 0.
The equation of the line is calculated same way as it is derived for rocket. However, the C.S.A. did not already know the equation of the line for meteor because they did not launch it. This time they actually had to find it themselves by charting the speeds and distances, and then finding the equation of the line by using one of the methods you learned of through this flyer. One of the things that alarmed the scientists at the C.S.A. the most when observing this meteor was its immense speed. Meteors generally enter the atmosphere at a speed of 600 km/m to 4320 km/m, but this meteor would enter the atmosphere at an approximate speed of 25,000 km/m!
The Solution of the Meteor and Rocket
You might have noticed, the C.S.A. calculated the line equation for the meteor and then they made sure that the rocket's line equation had a slope which is negative reciprocal of the meteor's slope. This is the fastest way for them to intercept.
While this may seem easy, do not underestimate the precision needed for this work. The C.S.A. had to plan very carefully, because if they did not, the meteor and rocket could pass by each other!
The Significance of the Solution
You now know:
- what the equation of a line is
- what the two types of standard form are
- multiple ways to find the equation of a line
- how to find the two types of standard form
Unfortunately we cannot educate you on how to make a rocket in a flyer, as it is much too complex, but who knows, with your newfound knowledge of linear systems, maybe one day of you will help intercept the next giant meteor that heads for Earth!