How Long Does It Take a Model Rocket to Reach Maximum Height?

AI Thread Summary
A model rocket is launched with an initial speed of 50.0 m/s and a constant upward acceleration of 2.00 m/s² until its engines stop at 150 m altitude. The correct method to determine the time to maximum height involves calculating the velocity at engine cutoff using kinematic equations, resulting in a total time of 8.52 seconds. An alternative approach attempted to calculate the time after engine cutoff but yielded incorrect results due to misapplication of equations. The confusion arises from the different methods producing varying values for time, highlighting the importance of consistent application of kinematic principles. Ultimately, the accurate calculation confirms the total time to reach maximum height is 8.52 seconds.
professordad
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Homework Statement



A model rocket is launched straight upward with an initial speed of 50.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s^2 until its engines stop at an altitude of 150 m. How long after lift off does the rocket reach the maximum height?

from College Physics by Serway and Faughn

Homework Equations



(1) v = v_0 + at where v denotes velocity, v_0 denotes initial velocity, a denotes acceleration, and t denotes time in seconds.

(2) x = v_0t + \frac{1}{2}at^2 where x denotes the displacement

(3) v^2 = v_0^2 + 2ax

The Attempt at a Solution



Ok so for this problem I found 2 ways to do it. One of them gives the correct answer (in the back of the book) and the other is off by a little.

The method that gives the correct answer:


So let the velocity of the rocket at the point where the acceleration stops be v_1 and the velocity of the rocket at the highest point be v_2. Let the time of the rocket's acceleration be t_1 and the time of no rocket's acceleration be t_2. Note that at the highest point, the velocity is 0, so v_2 = 0.

Now we find v_1. From equation (3), v_1^2 - 50.0^2 = 2 \cdot 2.00 \cdot 150 so v_1 = 55.68 m/s.

From equation (1), we have v_1 = 50.0 + at_1, so 55.68 = 50.0 + 2.00 \cdot t_1 and t_1 = 2.84 sec.

From equation (1) again, we have v_2 = v_1 + at_2, so -55.68 = -9.8t and t_2 = 5.68 sec.

The answer is \boxed{8.52} sec.


Ok so here's the other method that I tried but it didn't give the right answer:

Let d_2 denote the distance that the rocket travels after it stops accelerating itself. Then, by (3), 0 = 55.68^2 - 2 \cdot 9.8 \cdot x, so x = 158.

So by (2), x = 158 = 55.68 \cdot t_2 - 4.9t_2^2.

Solving the quadratic gives t_2 = 5.49 sec or t = 5.87 sec.

But this doesn't give the right answer :( , help?
 
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I get t=√3100/9.8=5.681 sec.
 
Yes :) because \sqrt{3100} = 55.68 as obtained in my first method, but why doesn't the second method work?
 
professordad said:
1. .


Ok so here's the other method that I tried but it didn't give the right answer:

Let d_2 denote the distance that the rocket travels after it stops accelerating itself. Then, by (3), 0 = 55.68^2 - 2 \cdot 9.8 \cdot x, so x = 158.

So by (2), x = 158 = 55.68 \cdot t_2 - 4.9t_2^2.

Solving the quadratic gives t_2 = 5.49 sec or t = 5.87 sec.

But this doesn't give the right answer :( , help?


I also wonder why 2 different values for t.
But if we put displacement=0 then,

0=55.68-4.9t
ttotal=55.68/4.9=11.3632
t=5.68163
 
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