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) =xe™.
(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 my = F)
(iii) Assume the particle starts in x9 = 0 with positive initial velocity vp > 0. Find the initial
energy Ep of the particle. Using (ii), show that the particle reaches x = 2 only if vo > 9, with
" 8e-2
t= —,
m
and in this case the particle’s velocity in x = 2 is
8e~2
v(2) = 4/ v2 — —.
(2)=8-—
(iv) Assume the particle starts in x9 = 0 with positive initial velocity vo > 9. Use (ii) to find the
expression for v(x) and find the terminal velocity of the particle as x — ce. If the particle starts
with negative initial velocity vo < 0, can it escape to x + —oo?
(v) Assume m = 1, show that the equation of motion is & =x(x—2)e™.
Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a
function of time f) for t € [0, 10] and for the six initial conditions
(a) x=0, vo =0
(b) x= 2, — =0
(c) x=0, v9 = 0.5
(d) x=0, v9 = —0.5
(e) x=0, v9 =2
(f) x =0, vo = —10.
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!)
,H1. A wooden block of mass m is pushed over a floor in a straight line in the x direction
with a constant velocity vg > 0. At time t = 0, when the block is at position x = 0, it is no
longer pushed and the block experiences only a constant sliding friction force F in the negative
x-direction until it comes to rest.
(i) Write down the equation of motion of the block for t > 0. Calculate the time, t,, at which
the block comes to a stop and also find the distance, d, travelled by the block from the time
when the pushing is stopped to the time when it comes to rest.
(ii) Calculate the work done by the friction force from x = 0 to x = d, and then show that it is
equal to the change in the kinetic energy of the block.