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Homework 5 solutions

# Homework 5 solutions - AMS 361 Applied Calculus IV by Prof...

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1 AMS 361: Applied Calculus IV by Prof Y. Deng Homework 5 Assignment Date: Wednesday (0 3 /11/2009) Collection Date: Wednesday (03/18/2009) Grade: Each problem is worth 10 points. Problem 5.1 (Prob. 27, P. 107, Modified) Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity 0 2 / v GM R . Then its height y(t) above the surface satisfies the initial value problem 2 2 2 ( ) d y GM dt y R   , (0) 0 y , 0 (0) y v . Compute the maximum altitude the projectile reaches and the time (from launch) for the projectile to reach the maximum altitude. Solution: The maximum altitude ( ? ?𝑎? ) : ? 2 ? ?? 2 = −𝐺? ( ? + 𝑅 ) 2 ; ? = ?? ?? ; ? 2 ? ?? 2 = ?? ?? ?? ?? = ? ( ?? ?? ) ? ?? ?? = −𝐺? ( ? + 𝑅 ) 2 ? ? 0 ?? = −𝐺? ( ? + 𝑅 ) 2 ?? ? 0 ? ( ? ) 2 2 ? 0 2 2 = 𝐺? ? ? + 𝑅 𝐺? 𝑅 => ? ( ? ) 2 = ? 0 2 2 𝐺?? ? 𝑅 ( ? ? + 𝑅 ) ? ? = 0 => ? ?𝑎? = 𝑅 ( 2 𝐺? 2 𝐺?−𝑅? 0 2 1) The time 𝑇 ?? (from launch) for the projectile to reach the maximum altitude: ?? ?? = ? = ? 0 2 2 𝐺?? ? 𝑅 ( ? ? + 𝑅 ) ,

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2 1 ? ? ?𝑎? 0 ?? = ?? 𝑇 ?? 0 , where T is the time the projectile going up.
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Homework 5 solutions - AMS 361 Applied Calculus IV by Prof...

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