Science:Math Exam Resources/Courses/MATH102/December 2017/Question 15 (b)

MATH102 December 2017

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[hide] Question 15 (b)

A grasshopper caught in a spiderweb rhythmically kicks its legs, causing the web to bounce up and down. The height of the grasshopper (in cm) after ${\textstyle t}$ seconds is $$h(t)=3+3\sin \left({\frac {5}{3}}t\right)$$ centimetres. (b) If the grasshopper can repeatedly reach a vertical acceleration of $5$ centimetres per second squared, it can free itself. Can the grasshopper free itself? (Remember to justify your answer.)

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!

[show] Hint
The acceleration is the second derivative of the displacement (height).

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[show] Solution

According to the Hint, let us compute the acceleration, which is the second derviative: applying the chain rule, we get $$h(t)=3+3\sin \left({\dfrac {5}{3}}t\right)$$ $$h'(t)=3\cos \left({\dfrac {5}{3}}t\right)\cdot {\dfrac {5}{3}}=5\cos \left({\dfrac {5}{3}}t\right)$$ $$h''(t)=-5\sin \left({\dfrac {5}{3}}t\right)\cdot {\dfrac {5}{3}}=-{\dfrac {25}{3}}\sin \left({\dfrac {5}{3}}t\right).$$ As ${\textstyle t}$ varies, we see that ${\textstyle h''(t)}$ oscillates between ${\textstyle -{\dfrac {25}{3}}}$ and ${\textstyle {\dfrac {25}{3}}}$. Since ${\textstyle {\dfrac {25}{3}}>5}$, the grasshopper can repeatedly (recall that the function ${\textstyle h}$ is periodic of period ${\textstyle {\dfrac {6\pi }{5}}}$ from part (a)) reach a vertical acceleration of ${\textstyle 5{\text{cm}}/{\text{m}}^{2}}$. So the grasshopper can free itself. Answer: ${\textstyle {\color {blue}{\mbox{Yes, since its vertical acceleration can reach up to }}{\dfrac {25}{3}}.}}$