# Derivatives : Velocity and Displacement

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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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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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.