Conclusions
Using the same (202,000,000 J) amount of mechanical impulse energy at the top or bottom of the ramp, impulse at the top of the ramp resulted in 204,080,604.5 J KE at the bottom of the ramp while impulse at the bottom of the ramp resulted in 260,921,883.60 J KE at the bottom of the ramp, which is more than the total mechanical impulse energy.
Even after returning to the surface on a second ramp, the “impulse at bottom vehicle” had more kinetic energy than the total energy in the mechanical impulse, while the “impulse at top vehicle” had less.
The impulse at top vehicle did worse in terms of efficiency than a standard car pushing off the ground after returning to the surface, while the “impulse on bottom” vehicle did better.
Vertical Drop Length: 4123.69 meters (29sec @ 1g - 2.56 miles)
Free-fall Velocity at Bottom of Ramp: 284.39 meters per second (636.16 miles per hour)
Water Tank Mass: 10000 kilograms
Passenger Vehicle Mass: 100 kilograms
Mechanical Impulse Energy: 202,000,000 J
Impulse on Top (202,000,000 J):
m1 = 100kg
m2 = 10000kg
v1 = 2000m/s
v2 = 20m/s
m1v1 = m2v2
v2 = (m1v1)/m2
v2 = (100kg*2000m/s)/10000kg
v2 = 20m/s
m1 KE = (1/2)mv^2
m1 KE = (1/2)100kg*2000m/s^2
m1 KE = 200,000,000 J
m2 KE = (1/2)mv^2
m2 KE = (1/2)10000kg*20m/s^2
m2 KE = 2,000,000 J
m1 KE + m2 KE = 200,000,000 J + 2,000,000 J = 202,000,000 J = Mechanical Impulse Energy
Covering 4123.69 meters with initial 2000 meters per second velocity and 1 g acceleration takes 2.07 seconds.
2.07 seconds with 1 g acceleration is a velocity change of 20.3 meters per second
m1 velocity at ramp bottom with impulse on top: 2020.3m/s = 2000m/s + 20.3m/s
m1 KE (bottom of ramp, impulse on top) = (1/2)mv^2
m1 KE (bottom of ramp, impulse on top) = (1/2)100kg*2020.3m/s^2
m1 KE (bottom of ramp, impulse on top) = 204,080,604.5 J
Impulse on Bottom (202,000,000 J):
Free-fall Velocity at Bottom of Ramp: 284.39 meters per second (636.16 miles per hour)
m1 = 100kg
m2 = 10000kg
v1 = 2000m/s
v2 = 20m/s
m1v1 = m2v2
v1 = (m2v2)/m1
v1 = (10000kg*20m/s)/100kg
v1 = 2000m/s
V1’s actual velocity is 2000m/s + the 284.39 freefall velocity at the bottom of the ramp:
2000m/s + 284.39m/s = 2284.39m/s
m1 KE (bottom of ramp, impulse on bottom) = (1/2)mv^2
m1 KE (bottom of ramp, impulse on bottom) = (1/2)100kg*2284.39m/s^2
m1 KE (bottom of ramp, impulse on bottom) = 260,921,883.60 J
Conclusions:
Max Velocity with no ramp: 2000m/s (4473.87mph @ ground)
Energy with 202,000,000 J Impulse on top: 200,000,000 J <----
Max Velocity “Standard Car” pushing off ground: 2009.97m/s (4496.19mph @ ground)
Energy with 202,000,000 J Impulse on top: 202,000,000 J <----
Max Velocity with impulse on top of ramp : 2020.3m/s (4519.28mph @ bottom)
Energy with 202,000,000 J Impulse on top: 204,080,604.5 J
Ground Level Exit Velocity with impulse on top of ramp: 2000.18m/s (4474.28mph @ ground)
Exit Energy with 202,000,000 J impulse on top of ramp: 200,036,002 J <----
Max Velocity with impulse at bottom of ramp : 2284.39m/s (5110.03mph @ bottom)
Energy with 202,000,000 J Impulse on bottom: 260,921,883.60 J
Ground Level Exit Velocity with impulse at bottom of ramp: 2266.64m/s (5070.33mph @ ground)
Exit Energy with 202,000,000 J impulse on bottom of ramp: 256,882,844 J <----
^Even after returning to the surface on a second ramp, the “impulse at bottom vehicle” had more kinetic energy than the total energy in the mechanical impulse, while the “impulse at top vehicle” had less.
The impulse at top vehicle did worse in terms of efficiency than a standard car pushing off the ground after returning to the surface, while the “impulse on bottom” vehicle did better.