Problem Set #1 SOLUTIONS
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Worcester Polytechnic Institute *
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Course
1110
Subject
Aerospace Engineering
Date
Feb 20, 2024
Type
Pages
3
Uploaded by DukeFang12576
PH1110 C23 Problem Set #1 Solutions Problem #1:
Consider the following equation: 𝑣(𝑡) = 𝑚𝑔
𝑏
(1 − 𝑒
−
𝑏
𝑚
𝑡
)
Where “v(t)” represents the instantaneous velocity as a function of time, “m” represents mass, “g” represents the acceleration due to gravity, “b” is a proportionality constant, and “t” is time. For this problem, it is not important to understand all aspects of the equation, but only to understand how units work within an equation and how to interpret the equation physically from the units. a. Through investigation of the given equation, what must be the units of “b,” the proportionality constant? Explain your answer. Given that the left hand side of the equation is velocity, which has SI units of meters/second as discussed in class, then “mg/b” as a whole must also have units of meters/second. If m = kilograms and g = meters/second/second, then we can set up a simple equation: ??????
??????
=
?𝒊??𝒈???? ∙
??????
??????
??????
?
Simplifying this then yields: 𝟏
𝟏
=
?𝒊??𝒈???? ∙
𝟏
??????
?
This then means that b = kilograms/second b. What would “mg/b” physically represent in this equation? Why? Explain your answer. As a hint, you may want to sketch the graphical trend to arrive at your answer. As a side note, I know some may be familiar with what it represents; however, I am looking for an explanation as to its physical significance. It must physically represent the maximum velocity that the system can achieve. If the equation is plotted, v(t) tends towards this value as time tends towards infinity. As a side note, this is also known as the terminal velocity, but knowing this term right now is not important. c. In the exponential term, what must be the units of “b/m”? Why? Exp
lain your answer. The units of b/m must be 1/seconds. This is because the exponential term cannot carry units, as it would be nonsensical to have a term where “e” is raised to some unit –
it would not be able to match on both sides. d. How long would it take for the object to reach one-fourth of its maximized velocity? Show all work! Rewrite the original equation as v = v
max
(1 –
e
-bt/m
), where v = v
max
/4. Thus: 1/4 = 1 –
e
-bt/m
3/4 = e
-bt/m t = ln(4/3)m/b
Problem #2:
The Mars Climate Observer mission was aimed at the study of what the mission’s name implies, and an orbiter was built to do the study (among other tasks). The cost associated with this mission in late 1990’s dollars was approximately $200 million (
from NASA
), which translates to $350 million in 2022 due to inflation. After a maneuver was performed to put the craft into orbit, no signal was received from Earth, thus resulting in a failure. It was discovered this was due to the failure to communicate units used in the burn process, where Lockheed Martin used English units (pound·seconds), whereas NASA used SI or metric units (Newton·seconds). Light mission details are given on NASA’s website (
here
), but you can also search for articles written at the time. This shows the importance of systems engineering! a. What is the conversion factor between pound·seconds and Newton·seconds? What do these quantities actually mean? You are encouraged to look these up –
I do not expect you to arrive at an answer on your own. 1 pound·
seconds ≈ 4.45 Newton
·seconds These units represent a term known as impulse, which is commonly used in rocketry. b. It was discovered that the unit analysis error occurred in the software that was meant to make course corrections. The software was reading out in the pound·seconds units, but other aspects of the craft meant to make corrections translated these to Newton·seconds. Were the numbers used in the course correction software too small or too large? What impact would this have? These numbers were too small by a factor of 4.45, bringing the orbiter far too close to Mars, which resulted in the craft crashing into the surface –
not the desired result of a stable orbit! Problem #3:
Refer to the diagram below: What is the magnitude and direction (measured relative to the +x axis) of the vector d
such that d
= a
+ b
+ c
?
Problem #4:
Refer to the diagram below: The figure shows a form that is to be used to pour a solid concrete wall. Note that two of the cables used to stabilize the form are connected to a common point P1 on the ground and run to opposite corners indicated by points P2 and P3. Dimensions are indicated in meters and note that the lower left edge of the form is at the origin of the indicated coordinate system. a. What is the vector from point P1 to point P2? b. What is the vector from point P1 to point P3? c. What is the angle between the two angles? As a hint, consider the definition of the dot product.
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