Science:Math Exam Resources/Courses/MATH102/December 2011/Question 08 (i)

MATH102 December 2011

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Question 08 (i)

You are driving down the highway when you see a sleeping moose. You apply the brakes and carefully stop your car 20m away from the animal. While you are looking for your camera the moose wakes up. It instantly charges toward your car at a constant speed of 8m/s. One second later, you start backing away from the moose at a constant acceleration of 2m/s2. (i) Write down a function d(t) that is the distance from your car to the moose where t = 0 indicates the moment when you start backing away.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
First, consider the displacement of the car and the moose separately. For example, fix the initial position to be where the car is stopped. Relatively to that point, what are the respective distance of the car and the moose over time. Then, you'll find the distance between the car and the moose by taking the difference of those two things.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $d(t)$ be the distance between your car and the moose, where t = 0 represents the time when your car begins backing up. The moose began charging one second before t=0, i.e. at t = -1, and so, when the car begins backing up the moose is 20m-8m/s*(1s) = 12m away. Let the origin be the position of the car at time t = 0. Consider the moose. If the moose is 12m away at t = 0 and is moving towards the car at a speed of 8 m/s, we can define the position of the moose as m(t) where $\displaystyle m(t)=12-8t.$ If we set $c(t)$ be the position of the car at time t. The car is accelerating away from the moose at 2 m/s2, then $\displaystyle c''(t)=-2\quad \rightarrow \quad c'(t)=-2t+c_{1}$ Since the car is not moving at t = 0, we have that $c'(0)=c_{1}=0$. Thus $c'(t)=-2t$. From this, we obtain that $c(t)=-t^{2}+c_{2}$. Since we set the position of the car at t = 0 to be $c(0)=0$, then $c(t)=-t^{2}$. The distance between the car and the moose is thus (note that the distance is always a positive number) $d(t)=|c(t)-m(t)|={\color {blue}|-t^{2}+8t-12|}$

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