A billiard ball (solid sphere of radius R) is struck such that at t = 0, it is strictly slipping with velocity v0. At some time t, it grows to roll without slipping. The coefficient of dynamic friction between the ball and the table is mu_k. a) Draw a free body diagram of all the forces acting on the billiard ball at t = 0. b) At what time t does the ball strictly roll? c) At what distance d from its initial position does this strict rolling occur?
a) Force of gravity acting down from center of mass, Normal force acting up from surface of the table, Force of friction acting in direction opposite motion from point of tangentiality between table and ball. b) t = v0/((7/2)*g*mu_k) c) d = (2*(v0)^2)/(49*mu_k*g)
A skydiver of mass m = 40 kg jumps out of a stationary helicopter at an elevation of H = 5 km. Air resistance of magnitude F = k*v acts on the skydiver during the descent, where v is the skydivers instantaneous velocity. a) Draw a free body diagram of forces acting on the skydiver. b) Write, but do not solve, a differential equation expressing the skydiver's motion. c) What is the skydiver's terminal velocity? d) Evaluate the differential equation from part b to find the skydiver's speed as a function of time. e) Calculate the skydiver's elevation as a function of time.
a) Force F applied upward, Force mg applied downward b) m(dv/dt) = mg - kv c) v = (mg)/k (occurs when dv/dt = a = 0) d) v(t) = (mg/k)(1 - exp(-kt/m)) (separation of variables to isolate dv and dt and integrate) e) x(t) = (mg/k)(t + (m/k)exp(-kt/m)) (integrate v(t)dt)
In December 2015, SpaceX's Falcon 9 rocket became the first to land its first stage on Earth after launch. Prior, all stage 1 rockets fell into the Atlantic Ocean, never to be recovered. To land successfully, the first stage had to be equipped with liquid rocket fuel capable of igniting both remotely and on command. The Falcon 9 liquid rocket engine and fuel weighs 5,000 kg and is capable of producing 75 kN of thrust. The engine is remotely set to fire for 60 seconds: a) What is the rocket's acceleration upward after ignition? b) What is the rocket's height h at the moment its engines stop firing? c) What is the rocket's maximum height H? d) What velocity vf should the rocket experience at the moment of landing? Why?
a) a = 5.19 m/s^2 (Newton's law in y direction) b) h= 9.342 km [ y(t) = .5*a*t^2 ] c) H = 14.28 km (H = h + delta_h, use conservation of energy to find delta_h after engine stops and add to h to find H, delta_h = v^2/2g) d) vf should equal 0 at landing so that the rocket experiences no impact force upon landing