H2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x)=x²e-x. (i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xe such that F(xe) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U" (xe) <0). Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in xo = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with 8e-2 v= m and in this case the particle's velocity in x = 2 is 8e- v(2)=√ v². (iv) Assume the particle starts in xo =0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x→∞. If the particle starts with negative initial velocity vo <0, can it escape to x → →∞? (a) x = 0, vo = 0 (b) x=2, vo=0 (c) x = 0, vo= 0.5 (d) x = 0, vo= -0.5 (e) x = 0, vo = 2 (f) x = 0, vo= -10. m (v) Assume m = 1, show that the equation of motion is d = x(x − 2)e¯*. Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a function of time t) for t = [0, 10] and for the six initial conditions Discuss the behaviour of the solutions in light of the previous points. (should you need help with the programming exercise, remember that weekly workshops are available for MATH1005 to help you set up your notebook file and get the code running!)
H2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x)=x²e-x. (i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xe such that F(xe) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U" (xe) <0). Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in xo = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with 8e-2 v= m and in this case the particle's velocity in x = 2 is 8e- v(2)=√ v². (iv) Assume the particle starts in xo =0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x→∞. If the particle starts with negative initial velocity vo <0, can it escape to x → →∞? (a) x = 0, vo = 0 (b) x=2, vo=0 (c) x = 0, vo= 0.5 (d) x = 0, vo= -0.5 (e) x = 0, vo = 2 (f) x = 0, vo= -10. m (v) Assume m = 1, show that the equation of motion is d = x(x − 2)e¯*. Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a function of time t) for t = [0, 10] and for the six initial conditions Discuss the behaviour of the solutions in light of the previous points. (should you need help with the programming exercise, remember that weekly workshops are available for MATH1005 to help you set up your notebook file and get the code running!)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![H2. A particle of mass m moves in a straight line under the action of a conservative force F(x)
with potential energy
U(x) = x²e-*.
(i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xẻ such that
F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy:
U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U"(xe) <0). Sketch the
potential energy as a function of x, indicating the equilibria on your plot.
(ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms
of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion.
(hint: use the equation of motion mv = F)
(iii) Assume the particle starts in xo =0 with positive initial velocity vo > 0. Find the initial
energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with
8e-2
v=
and in this case the particle's velocity in x = 2 is
v(2):
=
(a) x = 0, vo=0
(b) x=2, vo = 0
(c) x = 0, vo= 0.5
(d) x = 0, vo = -0.5
(e) x = 0, vo=2
(f) x = 0, vo= -10.
m
8e-2
v². m
(iv) Assume the particle starts in xo =0 with positive initial velocity vo > ŷ. Use (ii) to find the
expression for v(x) and find the terminal velocity of the particle as x → ∞. If the particle starts
with negative initial velocity vo < 0, can it escape to x →→∞?
(v) Assume m= 1, show that the equation of motion is d = x(x − 2)e¯x.
dt²
Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a
function of time t) for t = [0, 10] and for the six initial conditions
Discuss the behaviour of the solutions in light of the previous points.
(should you need help with the programming exercise, remember that weekly workshops are
available for MATH1005 to help you set up your notebook file and get the code running!)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed029406-a1c1-473f-a3a0-6fd0fbd8e89d%2Fcd4b5913-0149-422e-8a31-7da59654056c%2Ft2wpmhp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:H2. A particle of mass m moves in a straight line under the action of a conservative force F(x)
with potential energy
U(x) = x²e-*.
(i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xẻ such that
F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy:
U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U"(xe) <0). Sketch the
potential energy as a function of x, indicating the equilibria on your plot.
(ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in terms
of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion.
(hint: use the equation of motion mv = F)
(iii) Assume the particle starts in xo =0 with positive initial velocity vo > 0. Find the initial
energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with
8e-2
v=
and in this case the particle's velocity in x = 2 is
v(2):
=
(a) x = 0, vo=0
(b) x=2, vo = 0
(c) x = 0, vo= 0.5
(d) x = 0, vo = -0.5
(e) x = 0, vo=2
(f) x = 0, vo= -10.
m
8e-2
v². m
(iv) Assume the particle starts in xo =0 with positive initial velocity vo > ŷ. Use (ii) to find the
expression for v(x) and find the terminal velocity of the particle as x → ∞. If the particle starts
with negative initial velocity vo < 0, can it escape to x →→∞?
(v) Assume m= 1, show that the equation of motion is d = x(x − 2)e¯x.
dt²
Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a
function of time t) for t = [0, 10] and for the six initial conditions
Discuss the behaviour of the solutions in light of the previous points.
(should you need help with the programming exercise, remember that weekly workshops are
available for MATH1005 to help you set up your notebook file and get the code running!)
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