SMN 3043 1b

By gibsonsumbang11 | Updated: May 8, 2022, 11:20 a.m.

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Find 
b) The time required for the ball to hit the ground
x=3000 ft
We know that velocity is differential of position.

V = dr/dt
By integrating the equation from question 1(a), we can get the position formula using Ordinary Differential Equations

V(t) = 16 - 16 e

-2t
                                     ∫dx/dt =  ∫16 - 16 e

-2t
                                    x =  16t - (16 e   ) / - 2
-2t
                                    x =  16t + 8 e     + c
-2t
To find the constant, c ....
                                    0 = 16(0) + 8 e     + c
-2(0)
t = 0
                       c = - 8
Thus, the position of the steel ball is given by..
                                    x(t) =  16(t) + 8 e    - 8
-2t
                                    x(t) =  16(t) + 8 e    - 8
-2t
                                    3000 =  16(t) + 8 e    - 8
-2t
Substitute x = 3000 ft into equation

376 (8) =  (8)(2(t) +e    )  
                                    3008 =  16(t) + 8 e    
-2t
-2t
                                    376 = 2t +e     
-2t
Assume that   e  = 0 , since the equation cannot be solved explicitly.
-2t
                                    376 = 2t +e     
-2t
                                    376 = 2t  + 0    
2t = 376
  t = 188 s
Thus, the time required for the ball to hit the ground is 188 seconds.
t= 188 s
Thank you

SMN 3043 1b

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