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Derivatives : Velocity and Displacement

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27. A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building?

Answer: y0 = 178.57 m

35. A stone is dropped from rest at an initial height h above the surface of the Earth. Show that the speed with which it strikes the ground is v = √2gh.

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Derivatives are applied to velocity and displacement. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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27. A ball is thrown straight downward from the top of a tall building. The initial speed of the ball is 10 m/s. It strikes the ground with a speed of 60 m/s. How tall is the building?

Answer: y0 = 178.57 m

Solution. Assume that the height of the building is h meters. Denote by H(t) the height of a ball at time t. Note that the derivative of H(t) with respect to t is the speed; and the second derivative of H(t) is the acceleration of this ball, i.e., . Assume that the mass of a ball is M.

By the Newton's 3rd law, we have

i.e.,

Obviously, when t=0, H(0)=h and H'(0)=-10.

Note 1: H'(0) is the initial speed of the ball, so by given information, H'(0)= -10 (m/s)

NOTE 2: Why do we use -10 m/s? This is the direction of the ball is downwards!

So, we set up the ...

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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