Neglect the gravity of the Moon, neglect atmospheric friction, and neglect
the rotational velocity of the Earth in the following problem. A
long time ago, Jules Verne, in his book From Earth to the Moon (1865),
suggested sending an expedition to the Moon by means of a projectile
fired from a gigantic gun.
(a) With what muzzle speed must a projectile be fired vertically from a
gun on the surface of the Earth if it is to (barely) reach the distance of
the Moon?
Using the radius of the Earth and Moon as 3963 miles and 1080 miles, respectively, and the distance between them as 239,000 miles, the surface to surface distance becomes 233,960 miles.
From Vf^2 = Vo^2 - 2gs, 0 = Vo^2 - 2(32.2)233,960 or Vo = 282,053 fps = 192,309 mph.
The trip time then becomes t = 282,053/32.2 = 8760 sec = 2hr-26min.
Theoretically, the liftoff velocity should be 192,310 mph.
To find the muzzle speed required to reach the distance of the Moon, we need to consider the basic principles of projectile motion. Here are the steps to solve the problem:
Step 1: Gather relevant information
First, we need to find the distance from the Earth to the Moon. The average distance between the Earth and the Moon is approximately 384,400 kilometers.
Step 2: Determine the time of flight
Since we are neglecting the rotational velocity of the Earth and atmospheric friction, the only force acting on the projectile is gravity. The time of flight, t, can be found using the formula:
t = 2v₀/g
Where v₀ is the initial vertical velocity (muzzle speed) and g is the acceleration due to gravity, which is approximately 9.8 m/s².
Step 3: Calculate the vertical distance traveled
The vertical distance traveled by the projectile during the time of flight can be found using the formula:
d = (1/2)gt²
Since the projectile has to reach the distance to the Moon, the vertical distance should be equal to the distance between the Earth and the Moon.
Step 4: Solve for initial velocity (muzzle speed)
Rearrange the formula for vertical distance to solve for the initial velocity (muzzle speed):
v₀ = √(2gd)
Substitute the known values into the equation, with g = 9.8 m/s² and d = 384,400 km (converted to meters). Calculate the result to find the required muzzle speed to reach the Moon.
Keep in mind that this calculation neglects the gravity of the Moon and other important factors. In reality, reaching the Moon requires additional considerations and precise calculations.
Remember, this calculation assumes no air drag, no rotation of the Earth, and no gravity from the Moon.