Science:Math Exam Resources/Courses/MATH110/December 2017/Question 06 (b)

MATH110 December 2017

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Question 06 (b)

A small ball is attached to a vertical spring hanging from the ceiling. Suppose you pull the ball (vertically) down towards the ground. At ${\textstyle t=0}$ you let go and the ball begins to oscillate following the stretching and compression of the spring. Let $$h(t)=0.5\sin(2\pi (t-1))+2$$ be the position of the ball (in metres) at time ${\textstyle t}$ (in seconds) since it was released. Let the ground level be 0 m. (b) What is the velocity of the ball at ${\textstyle t=1}$?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that the velocity is the derivative of the displacement (position) function.

Hint 2
We need to find ${\textstyle h'(1)}$.

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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. By the Hints, we need to find ${\textstyle h'(1)}$. Using the chain rule with ${\textstyle f(t)=\sin(t)}$, ${\textstyle g(t)=2\pi (t-1)}$, ${\textstyle f'(t)=\cos(t)}$, ${\textstyle g'(t)=2\pi }$, we have $${\begin{aligned}h(t)&=0.5f(g(t))+2,\\h'(t)&=0.5f'(g(t))g'(t)\\&=0.5\cos(2\pi (t-1))\cdot 2\pi \\&=\pi \cos(2\pi (t-1)).\end{aligned}}$$ Therefore, $${\begin{aligned}h'(1)&=\pi \cos(2\pi (1-1))\\&=\pi \cos(2\pi \cdot 0)\\&=\pi \cos(0)\\&=\pi \cdot 1\\&=\pi .\end{aligned}}$$ Therefore the velocity at time ${\textstyle t=1}$ is ${\textstyle h'(1)=\pi }$. Answer: the velocity at ${\textstyle t=1}$ is ${\textstyle {\color {blue}\pi }}$ (metre per second).