Other Math Answers

Consider a new card game between 2 players: Darryl (player 1) and Phyllis (player 2)

Darryl is dealt two cards : ♡6 and ♠5. Phyllis is also dealt two cards: ♢3 and ♣9. Now, each of the players will play 1 card both at the same time.

The payoff of Darryl is 1 points if he plays a card of opposite color (red/black) than Phyllis, and otherwise his payoff is 6 points.

The payoff of Phyllis is 9 points if the difference of the already played card numbers is greater than 5, otherwise her payoff is 1 points.

Find the action sets of each player and the action profile of the game.

Represent the game in the Normal form.

Find the Best Responses for Darryl.

Find the Best Responses for Phyllis.

Find all the Nash Equilibriums of the game (if any).

A body is released from rest and moves under uniform gravity in a medium that exerts a resistance force proportional to the square of it's speed and in which the body's terminal speed is V. Show that the time taken for the body to fall a distance h is V/g cosh^-1e^gh/v .

In his famous (but probably apocryphal) experiment, Galileo dropped different objects from the top of the tower of Pisa and timed how long they took to reach the ground. If Galileo had dropped two iron balls, of 5mm and 5cm radius respectively, from a height of 25m, what would the descent times have been? Is it likely that this difference could have been detected? [Use the quadratic law of resistance with C=0.8. The density of iron is 7500kgm^-3.]

A particle is subject to uniform gravity and the linear resistance force -mKv, where K is a positive constant and v is the velocity of the particle. Initially the particle is projected with speed "\\mu" in a direction making an angle "\\alpha" with the horizontal.

(a). Find the vector equation of motion and the corresponding scalar equations of motion.

(b). Show that the solution for the trajectory of the particle at t=0,z=0,x=0 is x="\\mu"cos"\\alpha" /K (1-e^-Kt) z=K"\\mu"sin"\\alpha"+g/K² (1-e^-Kt)-g/K t

(c). Find an approximate formula for the range on level ground when the resistance parameter "\\lambda"=K"\\mu"/g is small.