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Ch 5.3 - 14E
Chapter 5, Problem 14E(choose chapter or problem)
(a) Use the substitution \(v=d y / d t\) to solve (13) for v in terms of y. Assuming that the velocity of the rocket at burnout is \(v=v_{0}\) and \(y \approx R\) at that instant, show that the approximate value of the constant c of integration is \(c=-g R+\frac{1}{2} v_{0}^{2}\).
(b) Use the solution for v in part (a) to show that the escape velocity of the rocket is given by \(v_{0}=\sqrt{2 g R}\). [Hint: Take \(y \rightarrow \infty\) and assume \(v>0\) for all time t.]
(c) The result in part (b) holds for any body in the Solar System. Use the values \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) and \(R=4000 \mathrm{mi}\) to show that the escape velocity from the Earth is (approximately) \(v_{0}=25,000 \mathrm{mi} / \mathrm{h}\).
(d) Find the escape velocity from the Moon if the acceleration of gravity is 0.165g and \(R=1080 \mathrm{mi}\).
Text Transcription:
v=d y/d t
v=v_0
y\approx R
c=-gR+frac12v_0^2
v_0=sqrt2gR
y\rightarrow\infty
v>0
g=32ftmathrms^2
R=4000mi
v_0=25,000mih
R=1080mi
Questions & Answers
QUESTION:
(a) Use the substitution \(v=d y / d t\) to solve (13) for v in terms of y. Assuming that the velocity of the rocket at burnout is \(v=v_{0}\) and \(y \approx R\) at that instant, show that the approximate value of the constant c of integration is \(c=-g R+\frac{1}{2} v_{0}^{2}\).
(b) Use the solution for v in part (a) to show that the escape velocity of the rocket is given by \(v_{0}=\sqrt{2 g R}\). [Hint: Take \(y \rightarrow \infty\) and assume \(v>0\) for all time t.]
(c) The result in part (b) holds for any body in the Solar System. Use the values \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) and \(R=4000 \mathrm{mi}\) to show that the escape velocity from the Earth is (approximately) \(v_{0}=25,000 \mathrm{mi} / \mathrm{h}\).
(d) Find the escape velocity from the Moon if the acceleration of gravity is 0.165g and \(R=1080 \mathrm{mi}\).
Text Transcription:
v=d y/d t
v=v_0
y\approx R
c=-gR+frac12v_0^2
v_0=sqrt2gR
y\rightarrow\infty
v>0
g=32ftmathrms^2
R=4000mi
v_0=25,000mih
R=1080mi
ANSWER:Step 1 of 5
In this problem we have to find the escape velocity of the moon.
(a)
Given that setting in the differential equation (13)
The differential equation become