* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download MATH10232: EXAMPLE SHEET X
Symmetry in quantum mechanics wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Routhian mechanics wikipedia , lookup
Fictitious force wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Hunting oscillation wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum tunnelling wikipedia , lookup
Path integral formulation wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Relational approach to quantum physics wikipedia , lookup
Atomic theory wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Grand canonical ensemble wikipedia , lookup
Four-vector wikipedia , lookup
Monte Carlo methods for electron transport wikipedia , lookup
Identical particles wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Mean field particle methods wikipedia , lookup
Particle filter wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Elementary particle wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Classical mechanics wikipedia , lookup
Equations of motion wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Centripetal force wikipedia , lookup
Brownian motion wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
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) 1 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 1 m|ṙ|2 + mgy = E, 2 and verify that the energy E is conserved during the motion. 1 Any feedback to: [email protected] 1 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) 2 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. 3 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! 4