2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. 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, in terms of vand 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 x0 = 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 > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 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 → ∞? (v) Assume m = 1, show that the equation of motion is d² = 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 (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 3 (f) x = 0, vo= -10. Discuss the behaviour of the solutions in light of the previous points.
2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. 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, in terms of vand 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 x0 = 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 > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 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 → ∞? (v) Assume m = 1, show that the equation of motion is d² = 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 (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 3 (f) x = 0, vo= -10. Discuss the behaviour of the solutions in light of the previous points.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...
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![2. A particle of mass m moves in a straight line under the action of a conservative force F(x)
with potential energy
U(x) =
= x²ex.
(i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi-
libria are stable or unstable. 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, in terms of vand 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 x0 = 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 > ô,
with
8e-2
v =
m
and in this case the particle's velocity in x = 2 is
8e-2
༧(2)
vo
m
(iv) Assume the particle starts in x0 = 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 → ∞?
(v) Assume m = 1, show that the equation of motion is d² = 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
(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
3
(f) x = 0, vo= -10.
Discuss the behaviour of the solutions in light of the previous points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F551b2f87-bba7-4106-b9d6-edcf3e523c90%2Fd9dec067-72df-411f-bf52-d79abce1c255%2Fi588auk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. A particle of mass m moves in a straight line under the action of a conservative force F(x)
with potential energy
U(x) =
= x²ex.
(i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi-
libria are stable or unstable. 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, in terms of vand 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 x0 = 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 > ô,
with
8e-2
v =
m
and in this case the particle's velocity in x = 2 is
8e-2
༧(2)
vo
m
(iv) Assume the particle starts in x0 = 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 → ∞?
(v) Assume m = 1, show that the equation of motion is d² = 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
(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
3
(f) x = 0, vo= -10.
Discuss the behaviour of the solutions in light of the previous points.
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