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Unformatted text preview: PHYS 3113 – Analytical Mechanics Fall 2011 Exam 3 Problem 3.1 (35 pts) A particle moves in a potential V ( r) = − γ2 where γ is a constant and r is the distance r
from the origin. a) Calculate the force on the particle. b) Write the differential equation of the orbit. €
c) Assuming 2γ/ml2 < 1, solve the differential equation for the orbit. Use A and θ0=0 as your constants of integration. d) Compute θ(t). Problem 3.2 (30 pts) An artillery shell is fired at an angle of 45° above the horizon with an initial speed of v0. At the uppermost part of its trajectory, the shell bursts into two equal fragments, one of which moves directly downward, relative to the ground, with an initial speed of v0/2. What is the direction and speed of the other fragment immediately after the burst? Problem 3.3 (40 pts) Consider a rocket with initial mass m0, taking off vertically from rest in a gravitational field g, so that the equation of motion becomes: −mg = mv + v m
˙ ex ˙ where vex is the speed of the rocket exhaust. Assume that the rocket ejects mass at a constant rate, –k, where k is a positive constant, so that m = m0 – kt. a) Solve for the velocity, v, as a function of time (remember, m is a function of t). €
b) Now integrate v(t) and show that the rocket’s height as a function of t is: 1
mv ex m y ( t ) = v ex t − gt 2 +
ln . 2
k m0 c) If the initial mass of a space shuttle is 2 x 106 kg and after about 2 minutes after takeoff the mass is 1 x 106 kg, and the exhaust speed is about 3000 m/s, estimate the shuttle’s height after 2 minutes. € Note: ∫ (ln u) du = u ln u − u + C € € 1 ∫ cos 2 u du = 2 ∫ (sec u)du = tan u + C ...
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