Maximum altitude of a rocket engine

AI Thread Summary
The discussion centers on calculating the maximum altitude, total time of flight, and horizontal range of a rocket launched at an angle of 53 degrees with an initial speed of 100 m/s and an acceleration of 30 m/s² for 3 seconds before engine failure. The correct maximum altitude is noted as 1521.5 m, total time of flight as 36.1 s, and range as 4045 m, but the original poster's calculations yield different results. Key errors identified include not accounting for the vertical component of the rocket's initial acceleration in the altitude calculation and neglecting the horizontal component of acceleration in the range calculation. Suggestions include using the correct trajectory equations to resolve these discrepancies. Accurate application of these principles is crucial for obtaining the correct answers.
imatreyu
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Homework Statement


A rocket is launched at an angle of 53 above the horizontal with an initial speed of 100 m/s. It moves along its initial line of motion with an acceleration of 30 m/s^2 for 3 s. A this time its engines fail and the rocket proceeds to move as a free body.

a. Find maximum altitude. =1521.5 m (Correct answer, according to some sources. . .)
b. Its total time of flight = 36.1 s (Correct answer, according to some sources. . .)
c. It's range =4045 m (Correct answer, according to some sources. . .)

I have completed this problem, however my answers do not seem to agree with what the correct answers are supposed to be. The correct answers are noted above. Could someone please point out where I went wrong? I feel like I understand why my process should work, but it's clearly wrong. . .

The Attempt at a Solution


a.
-The engine stops working at this vertical height: dy =100 sin 53 (3) + 1/2 (30) (3^2)=374.590653
-The rocket's speed after the engines fail is: vf= vi + at = 100 + 30 (3) = 190 m/s
-When the rocket fails, it is still moving at the original angle. Resolved, the vertical velocity is: viy= 190sin53

So: vf^2= vi^2 - 2gd ---> d = -vi^2 / (-2 (9.8)) = 1174.757871

Then, to find the maximum altitude: 1174.757871 + 374.590653 = 1549.35 m

b.
-Time with the engine working is given = 3s.

-Time w/o the engine: vf = vi +at --> 0 = 190sin53 -9.8t --> t= 15.48374968s

-Time to fall: vf= vi + at --> t= vf-vi/a--> t= (squareroot 2ad )/a --> t= (squareroot 2 * 9.8 * 1549.35)/ 9.8 = 17.78184123 s

Adding all three values together = 36.26559091 s.c.
There are two different horizontal velocities: 100cos53 and 190cos53. I solve for both:

For the acceleration period, the horizontal range should be:
dx= 100cos53 (3) + 1/2 (30) 3^2
dx= 315.5445069

For the time after the engines fail:
dx= 190cos53 (33.26559091) + 0 (there is no horizontal acceleration after the engines fail)

Added together, the two values give a horizontal range of= 4119.293656 m.Thank you in advance! I very much appreciate any help given to identify what I am doing wrong!
 
Physics news on Phys.org
In First Part you have not taken component of rocket's initial acceleration in y direction
dy =100 sin 53 (3) + 1/2 (30) (3^2)sin53 = 347.406
 
In Third Part you have not taken component of rocket's initial acceleration in x direction
When engine fails at this point use
equation of trajectory
y=xtanA-(gx^2)/(2u^2Cos^2A)
A= 53
 
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