February 19th, 2012, 01:32 PM  #1 
Newbie Joined: Feb 2012 Posts: 5 Thanks: 0  Falling Object
1. If a ball is thrown vertically upward from the roof of a 32 foot building with a velocity of 32 ft/sec, then what is the maximum height the ball reaches? 2. What is the velocity of the ball when it hits the ground? Can someone please help me? I was never taught the formula or how to solve this kind of problem... And I'd be grateful to anyone that can help me xP And if you can... use the most simple words you can to explain this OTL Yeah, I'm dumb. Sorry 
February 19th, 2012, 02:07 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Falling Object
There are two ways we can solve these questions: dynamics and conservation of energy. Dynamics: Once the ball is thrown, if we ignore drag, the only force working on the ball is that of gravity, so the acceleration of the ball is down with a magnitude of g. We can use the fact that acceleration is defined to be the time rate of change of velocity, and the given initial values to set up the initial value problem: where We also have the definition that velocity is the time rate of change of position, or . We may remove time from the equation by using the chain rule as follows: where Now, integrating with respect to x, using boundaries as limits, we find: Now, when the object has reached its maximum height, its velocity is zero, thus we then have: Now, it's just a matter of using the given data to find x. To find the velocity when the object hits the ground, set x to the value of the ground and solve for v: We tAke the negative root since the object is moving down: Now, it's just a matter of using the given data to find v. I have to run now, but when I get back I will show how to use energy to solve these as well. 
February 19th, 2012, 06:31 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Re: Falling Object Hello, Ahsayuni! Quote:
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February 19th, 2012, 06:48 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Falling Object
If we orient the axis of motion, the yaxis this time such that up is in the positive direction and the origin is at the ground, then the initial energy of the ball is part gravitational potential and part kinetic: When the ball reaches its maximum height, all of its energy is gravitational potential, but equal to the initial energy: As you can see, this is the same result as in my first post. When the ball reaches the ground, all of its energy is kinetic energy, but equal to the initial energy: Again, taking the negative root since the ball is moving down: This is also equivalent to the result in my first post, given that . 
February 20th, 2012, 06:07 PM  #5 
Newbie Joined: Feb 2012 Posts: 5 Thanks: 0  Re: Falling Object
Thank you everyone that replied to my post! You've all been a great help @_____@ I finally understand how to do falling object problems now >__< Thanks~ 
February 20th, 2012, 06:10 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Falling Object
Glad to be a part of the help, and welcome to the forum! 

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