Solved by Angela M K Foudray

Question:

Plutoball

We are playing baseball on an airless planet. A fly ball is hit directly towards an outfielder. The fielder can detect the changing angle that the ball makes with the horizon, but cannot directly judge changes in distance (apparent size of the ball), and there is no apparent lateral motion. Is there a way for the fielder to determine whether to move forward or backward to catch the ball? Prove your answer.

Answer:

Known:

theta(t) : the angle the ball makes wrt the ground at all times

a : the gravitational constant of the planet

t : how much time has gone by since the ball was

hit (watch with millisecond precision is a requirement for all ball

players)

If the players are playing on this planet for the first time, they

can find out what the gravitational constant of the planet is before

the game by knowing the height of one of their players and measuring

the time it takes to hit the ground.

h = 0.5 * a * t^2 (h is the height of the player, t is the time

it took to hit the ground from being dropped from rest from his

height)

Now, back to the game:

This is an airless planet => no drag => velocity parallel to the

ground in direction towards outfielder is constant!

The time for the ball to drop from h to the height it was hit

from is the same as the time from the hit to the max height, h.

This time, because ball players know their physics, is the same time

it takes for the ball to simply drop from rest from the height h to

the height it was hit at (the velocity at the peak of the parabola

is all in the direction towards the outfielder - that vector

component is zero, so the initial conditions are the same).

h = 0.5 * a * (t_1/2)^2 (h is the max height, a is the known

gravitational acceleration, and t_1/2 is half the total flight time

time from hit to peak height)

now we know h.

Conserving energy,

y  =  v_y * t  +  0.5 * a * t^2

(assuming the height it was hit is the same height it's caught)

0  =  (-v_i)_y * 2 * t_1/2  + 0.5 * a * (2 * t_1/2)^2

(where -(v_i)_y is the negative of the initial velocity in

the y direction)

(v_i)_y  =  v_i * sin(theta_i)  =  0.5 * a * 2 * t_1/2  =>

(where theta_i is the initial angle the ball was hit at wrt

the ground)

v_i * sin(theta_i)  = a * t_1/2  =>  v_i  =  (a * t_1/2) / 

sin(theta_i)

and (v_i)_x  =  v_x  =  v_i * cos(theta_i) (v_x

is a constant because of the lack of air)

and x  =  v_x * (2 * t_1/2)

There you go!  He can calculate the position he needs to stand in to

catch the ball!

-Ange