Download MATH10232: EXAMPLE SHEET X

MATH10232: EXAMPLE SHEET1 X
Questions for supervision classes
Please hand in answers to questions 2 and 3,
but attempt all questions.
1. Projectile motion
A particle P of constant mass m has position
r(t) = x(t) i + y(t) j,
where i and j are the base vectors of a global Cartesian coordinate
system in an inertial frame of reference. The particle is influenced by
a uniform gravitational field −gj. At time t = 0, the particle is at the
origin of the coordinate system and is projected with speed U at an
angle 0 ≤ θ ≤ π to the base vector i.
(a) Use Newton’s second law of motion to show that
x(t) = Ut cos θ,
(b)
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y(t) = Ut sin θ − gt2 .
2
i. Find the time τ at which the particle returns to ground level
(y = 0).
ii. Find the distance R of the particle from the origin at time τ .
iii. For fixed U, find the angle θ that maximises the distance R.
(c) Starting from Newton’s second law, derive the energy equation
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m|ṙ|2 + mgy = E,
2
and verify that the energy E is conserved during the motion.
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2. Firing projectiles in the wind
The particle described in question 1 is again projected from the origin
at time t = 0 with speed U at an angle 0 ≤ θ ≤ π to the base vector i.
A stiff wind has started to blow and imposes a force mγ (W − v) on
the particle, where v is the particle’s velocity, W is the wind velocity
and γ > 0 is a constant.
(a) Use Newton’s second law to show that the velocity of the particle
satisfies the vector-valued ODE
v̇ + γv = γ W − g j.
(1)
(b) Show that the general solution of equation (1) is
v = A e−γt + W −
g
j,
γ
where A is a constant vector.
(c) Hence, show that
g
1
g
U − W + j 1 − e−γt ,
r = W − j t+
γ
γ
γ
where U = U cos θ i+U sin θ j is the initial velocity of the particle.
(d) If the wind blows with speed W at an angle φ to the base vector i
show that that the time τ at which the particle returns to ground
level (y = 0) satisfies the equation
(1 − e−γτ ) =
(g − γW sin φ) τ
.
U sin θ − W sin φ + g/γ
Explain why a purely horizontal wind (acting parallel to the idirection) does not affect the time taken for the particle to return
to ground level.
(e) If the wind is purely horizontal show that the horizontal displacement of the particle from the origin is given by
R=
W sin θ + (g/γ) cos θ
τ.
sin θ + g/(γU)
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3. Motion in potential wells
A particle of unit mass moves along the positive x-axis under the influence of a force F (x) = F (x) i, where i is a unit vector in the direction
of increasing x.
2a
bn
F (x) = 3 − n+1 ,
x
x
(a) Find and sketch potential V (x) for the three cases
i. a = 1, b = 5, n = 1.
ii. a = −1, b = 5, n = −1.
iii. a = 1, b = −5, n = −1.
Henceforth, consider only the case a = 1, b = −5, n = −1.
(b) Show that the energy of the particle E ≥ E0 , where E0 is a constant to be determined.
(c) A particle is released from rest at the point x = 1, show that the
particle oscillates between the two points x = 1 and x = α, where
α is to be determined.
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4. Computer Exercises
The aim this week is to explore one-dimensional motion in potential
wells with MATLAB.
(a) Write a MATLAB function
function potential(V,x_init,v_init)
that
• Plots a graph of the potential V (x) over the range 0.1 ≤ x ≤
10, say.
• Simulates the motion of a particle of unit mass released from
the point x_init with initial velocity v_init.
You can use Euler’s method to solve the momentum equation
mẍ = F,
using a finite-difference approximation to find F = −V ′ (x),
F (x) ≈ −
V (x + ǫ) − V (x)
,
ǫ
where ǫ = 10−8 , say.
(b) Simulate the motion corresponding to question 3(c) and check that
the particle oscillates between x = 1 and x = α, as predicted. If
you use too large a timestep then you will find that energy is not
conserved!
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