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DYNAMICS - EXAMPLE


Toy Rocket
  Example

 

A small toy rocket follows a vertical path that can be modeled by y(t) = 5t3/6 - t4/12 (ft). What will be the maximum height? Also, what is the height when the acceleration is zero?

 

   
    Solution


Position when v = 0 and a = 0

 

First, find the equation that models both the velocity and acceleration as function of time.

     v(t) = dx/dt = 5t2/2 - t3/3

     a(t) = dv/dt = 5t - t2

a) The maximum height will occur when velocity is zero (it has reached it zenith).

     v = 0 = 5t2/2 - t3/3
     t = 7.5 s

Substitute time back into the position equation gives,

     yt=7.5 = 5 (7.5)3/6 - (7.5)4/12
              = 87.89 ft

b) Use the acceleration equation to find the time when the acceleration will be zero.

     a = 0 = 5t - t2
      t = 5 s (zero is not a realistic solution)

Substitute time back into the position equation gives,

     yt=5.0 = 5 (5)3/6 - (5)4/12
              = 52.08 ft

So, the rocket has a negative acceleration after the rocket goes up only 52.08 ft, but it keeps climbing until it reaches its zenith of 87.89 feet.

     
   
 
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