October Sky#
Day 1: Projectile Motion and Impulse#
Problem 1
In October Sky, the boys build small rockets and test them repeatedly. Suppose one rocket is launched from ground level with speed \(v_0 = 52\ {\rm m/s}\) at an angle \(\theta = 58^\circ\) above the horizontal. Take \(g = 9.81\ {\rm m/s^2}\). Neglect air resistance and assume the rocket lands at the same height from which it was launched.
(a) What are the horizontal and vertical components of the launch velocity?
(b) How long is the rocket in the air?
(c) What horizontal range does the rocket travel?
Problem 2
A rocket motor produces thrust by ejecting gas backward. Suppose a small rocket of mass \(m = 0.85\ {\rm kg}\) experiences an upward thrust of \(F_T = 28\ {\rm N}\) during powered flight. Take \(g = 9.81\ {\rm m/s^2}\).
(a) What is the weight of the rocket?
(b) What is the net upward force on the rocket?
(c) What is the rocket’s upward acceleration? Assume the mass remains constant during this short interval.
Problem 3
When a rocket is launched, the impulse from the engine changes the rocket’s momentum. Suppose a rocket of mass \(m = 0.75\ {\rm kg}\) starts from rest and reaches speed \(v = 38\ {\rm m/s}\) after leaving the launcher.
(a) What is the rocket’s final momentum?
(b) What impulse was delivered to the rocket?
(c) If the thrust acts for \(\Delta t = 1.6\ {\rm s}\), what is the average net force on the rocket?
Day 2: Energy and Aerodynamics#
Problem 4
A rocket launched straight upward reaches a maximum height before falling back down. Suppose a rocket leaves the launcher with vertical speed \(v_o = 64\ {\rm m/s}\). Take \(g = 9.81\ {\rm m/s^2}\) and neglect air resistance after burnout.
(a) What maximum height does it reach above the launch point?
(b) How long does it take to reach that maximum height?
Problem 5
The boys must make their rockets stable in flight. One factor is the drag force on the rocket body. Assume a rocket has cross-sectional area \(A = 3.5\times 10^{-3}\ {\rm m^2}\), drag coefficient \(C_D = 0.75\), air density \(\rho = 1.20\ {\rm kg/m^3}\), and speed \(v = 45\ {\rm m/s}\). Use \(F_D = \frac{1}{2}C_D \rho A v^2.\)
(a) What drag force acts on the rocket?
(b) If the rocket mass is \(0.90\ {\rm kg}\), what deceleration would this drag force alone produce?
(c) Explain why improving rocket shape can increase the height reached.
Problem 6
One way to estimate rocket performance is to compare launch kinetic energy to gravitational potential energy at maximum height. Suppose a rocket of mass \(m = 0.80\ {\rm kg}\) leaves the launcher at \(v_0 = 55\ {\rm m/s}\) and rises to a maximum height of \(h = 130\ {\rm m}\).
(a) What is the rocket’s initial kinetic energy?
(b) What is its gravitational potential-energy increase at the top?
(c) What fraction of the initial kinetic energy remains after accounting for the gain in gravitational potential energy?
(d) Where did the rest of the energy likely go?