Answer to Question #2774 in Abstract Algebra for carson

Question

Suppose two gliders start at the same height, one with a glide ratio of 0.3 and one with a glide ratio of 2/7. If they glide until they hit the ground, which one will have glided the farther horizontal distance?

Answer 1

Let u=(ux,uy) be the coordinates of the initial velocity of the first glider, and v=(vx,vy) be the coordinates of the initial velocity of the second glider.
Then by the definition glider ratio is the quotient:
ux / uy = 0.3,
vx / vy = 2/7

The move of each glider can be regarded as a sum of two moves: vertical and horizontal.
The vertical move has negative acceleration
a = -9.8 m/s^2, with initial velocity uy,
while the horizontal move has constant velocity ux.
Similarly for the move of the second glider.

Since& they hit the ground at the same moment T, starting from the same height, it follows that they come the same paths:
Sy1 = uy * T& - g T^2 /2
Sy2 = vy * T& - g T^2 /2
As& S1=S2 and t is the same, it follows that uy=vy.
Denote this number by& U.

Then& ux / U = 0.3, and thus
ux = 0.3 U,
and similarly
vx=2/7 U.

Since the horizontal moves have constant velocities, we have that
Sx1 = ux * T = 0.3 * U * T = 21/70 * U * T
Sx2 = vx * T = 2/7 * U * T = 20/70 * U * T,
so
& Sx1 > Sx2

Hence the first glider come the farther horizontal distance.