Problem Set 2 (Given 9/25/11)

Level I

Buford the bully grabs Baljeet's calculator and throws it upward at velocity \(v_0 = 4 \frac{m}{s}\), from a height of \(1m\) (his height). How much time does Baljeet have before the calculator hits the ground?
Isabella and Buford are competing to see who can throw a baseball farthest. Isabella throws the ball at \(v_i = 4\frac{m}{s}\) and Buford at \(v_b = 5\frac{m}{s}\); but Isabella throws the ball at a 45 degree angle, while Buford chucks it at a 60 degree angle from the ground. Who wins? (Assume the ball is released from ground level.)
A circular platform of radius \(10m\) rotates at an angular velocity of \(4\frac{1}{s}\). At the edge of the platform is a smaller circular platform, of radius \(1m\), rotating at an angular velocity of \(3 \frac{1}{s}\). From the point of view of someone no on the platform, at what velocity is the edge of this smaller platform furthest from the larger platform's center moving?
Perry the Platypus falls from a height of \(h = 500m\). He's struggling to open a jammed parachute, opening which would make him decelerate at \(4 \frac{m}{s^2}\). (That is, instead of accelerating at \(-g\), his acceleration would be \(4 \frac{m}{s^2}\), where positive is upwards.) How much time does he have to unjam it if he wants to hit the ground with no velocity? ¹ [¹ This is actually a terrible model for parachutes.]

Isabella and Buford are back for a rematch. This time they are on a platform above ground level, so it's not clear what the best angle to throw the ball is. Isabella throws it at 30 degrees, again at \(v_i = 4\frac{m}{s}\), and Buford throws it at 65 degrees, at \(v_b = 5\frac{m}{s}\). How much above ground level does the ball have to be released for one or the other to win? What if they instead start in a pit — how deep must it be for each to win?
Two trains approach each other at speed \(v\). A bird alights off the front of one train and starts toward the other at a speed \(2v\). When it reaches the other train it turns around and flies back, repeating until it gets crushed. Just as the bird leaves the train for the first time, a bee alights off its beak, traveling at \(3v\), and flies back and forth between the bird and the train until the two trains crash. How much total distance did the bee cover?
Perry the Platypus jumps from a height of \(50m\). When he is \(25m\) above the ground, he fires his grappling hook gun at a ledge \(500m\) up. What speed must the grappling hook gun fire grappling hooks for Perry not to crash into the ground?

A ball is attached by a \(1m\) rope to the origin, and in a circle spins about it at some angular velocity \(\omega\). Determine values of \(\omega\) such that there is some point at which you could cut the rope so that the ball would pass through the point \((5m, 5m)\) as it flies off. Consider positive and negative values of \(\omega\).
A piece of gum is stuck to a wheel of radius \(r\). The wheel is rolling around a cylinder of radius \(R\) (you might imagine a wheel rolling around the earth) at angular velocity \(\Omega\) (this is the angular velocity of the center of the wheel with respect to the center of the cylinder). Find the equations of motion of the piece of gum.

This is actually a terrible model for parachutes.