A projectile is launched vertically from the surface of the Moon with an initial speed of 1400 .
At what altitude is the projectile's speed one-half its initial value?
do you use the equation Ei = Ef and vf would be 700
E= 1/2 mv^2 -G (Mass of the moon/radius)
To find the altitude at which the projectile's speed is one-half its initial value, let's use the principle of conservation of mechanical energy.
The equation you mentioned, Ei = Ef, represents the conservation of mechanical energy. In this case, Ei represents the initial mechanical energy and Ef represents the final mechanical energy.
The initial mechanical energy (Ei) of the projectile can be expressed as the sum of its kinetic energy (KEi) and potential energy (PEi):
Ei = KEi + PEi
Since the projectile is launched vertically, the initial potential energy is given by PEi = mgh, where m is the mass of the projectile, g is the acceleration due to gravity, and h is the initial altitude.
The initial kinetic energy (KEi) is given by KEi = 1/2 * m * v^2, where v is the initial speed of the projectile.
At the point where the projectile's speed is one-half its initial value, the final kinetic energy (KEf) is given by KEf = 1/2 * m * (v/2)^2 = 1/8 * m * v^2, assuming the mass of the projectile remains constant.
Now, let's set up the conservation of mechanical energy equation:
Ei = Ef
KEi + PEi = KEf
1/2 * m * v^2 + m * g * h = 1/8 * m * v^2
Simplifying the equation:
4 * (1/2 * m * v^2) + 8 * m * g * h = m * v^2
2 * m * v^2 + 8 * m * g * h = m * v^2
6 * m * g * h = m * v^2
Divide both sides of the equation by m * v^2:
6 * g * h = v^2
Now, substitute the given values into the equation:
6 * (acceleration due to gravity on the Moon) * h = (initial speed)^2/4
Solving for h:
h = (initial speed)^2 / (24 * g)
Substituting the values, with the initial speed = 1400 m/s and the acceleration due to gravity on the Moon being approximately 1.622 m/s^2:
h = (1400^2) / (24 * 1.622)
h ≈ 140,697 meters
Therefore, at an altitude of approximately 140,697 meters, the projectile's speed will be one-half its initial value.