Science:Math Exam Resources/Courses/MATH100 A/December 2024/Question 11 (c)

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH100 A December 2024

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q10 (a)  •  Q10 (b)  •  Q11 (a)  •  Q11 (b)  •  Q11 (c)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q13  •  Q14  •  

   Other MATH100 A Exams

  •  December 2023  •  December 2024  •  

Question 11 (c)

A weight on the end of a spring is placed in a viscous liquid (e.g. olive oil). The position of the object in the liquid at time $t$ seconds is $x(t)$ metres. At time $t=0$, the weight is held still at position $x(0)=2$ with velocity $v(0)=0$. It is then released and starts moving so that the position is $$x(t)=e^{At}(B\sin (t)+C\cos (t))$$ where $A,B,C$ are constants. After a very long time, the weight seems to stop moving. Find all values of the constant $B$ that are consistent with the given information.

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
Use the initial conditions $x(0)=2$ and $v(0)=0$. What relationships between the constants $A,B,\text{and}C$ are implied by these conditions?

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
We are told that $v(0)=x^{\prime }(0)=0$. We also have that $$v(t)=x^{\prime }(t)=A{e}^{At}(B\sin (t)+C\cos (t))+e^{At}(B\cos (t)-C\sin (t)),$$ and hence, $0=v(0)=AC+B$. We also know from part (a) that $C=2$, so $B=-2A$. Finally, from part (b) we know that $A$ is negative, so we must have that $B>0$.

{{#incat:MER CT flag||

Click here for similar questions

MER CT flag, MER QGH flag, MER QGQ flag, MER QGS flag, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

}}