Rоcket Burn аnd Mаximum Altitude PHY 2048C Cumulаtive Final Examinatiоn Pоints: 10 Suggested Time: 25–30 minutes Instructions: Show all work. Begin each derivation with an appropriate fundamental physics principle. Clearly define any additional symbols you introduce. Unsupported answers may not receive full credit. A research rocket launches vertically from rest. During the powered portion of the flight, fuel is expelled downward at a constant speed u relative to the rocket. Fuel is consumed at a constant rate λ, so the rocket's mass is m(t) = m0 − λt, 0 ≤ t ≤ tb where m0 is the initial mass and tb is the burnout time. Assume m0 > λtb. Neglect air resistance, take upward as positive, and treat g as constant. After burnout, the rocket coasts upward under gravity alone. Tasks Part A. Starting from conservation of momentum for a variable-mass system, derive the rocket's velocity v(t) during the burn. Your final expression must be in terms of u, m0, λ, t, and g. Part B. Determine the rocket's burnout speed vb. Part C. Derive an expression for the vertical distance yb traveled during the powered portion of the flight. You may use the initial conditions y(0) = 0 and v(0) = 0. Part D. Determine the rocket's maximum altitude ymax above the launch point. Part E. State the minimum condition on u, λ, m0, and tb required for the rocket still to be moving upward at burnout.
Pulley System with Rоtаtiоnаl Inertiа PHY 2048C Cumulative Final Examinatiоn Points: 10 Suggested Time: 20–25 minutes Instructions: Show all work. Begin each derivation with an appropriate physics principle. Clearly define any additional symbols you introduce. Block 1 of mass m1 rests on a frictionless incline of angle θ. It is connected by a light cord over a pulley to a hanging block of mass m2. The pulley has radius R and moment of inertia I. The cord does not slip on the pulley. Assume block 2 moves downward and block 1 moves up the incline. Tasks Part A. Write Newton's second-law equation for each block and the rotational equation for the pulley. Part B. Derive the magnitude of the acceleration a of the system. Part C. Determine the two cord tensions T1 and T2. Part D. Starting from rest, determine the speed v of the blocks after block 2 descends a distance s. You may use either Newton's laws or conservation of energy.