A projectile is fired vertically from Earth's surface with an initial speed v0. Neglecting air drag, how far above the surface of Earth will it go? (Use any variable or symbol stated above along with the following as necessary: ME for the mass of the Earth, rE for the radius of the Earth, and G for the gravitational constant.)
h =
dum af
The equations of accelerated motion, derived elsewhere, also apply to falling (or rising) bodies with the exception that the term "a" for acceleration is replaced by the term "g” for the acceleration due to gravity). This results in
....Vf = Vo + gt (the term “g” for acceleration due to gravity is assumed constant)
....d = Vo(t) + g(t^2)
..................2
....Vf^2 = Vo^2 + 2gd
where Vo = the initial velocity, Vf = the final velocity, g = the acceleration due to gravity, t = the time elapsed and d = the distance covered during the time period t.
g derives from g = GM/r^2 where
G = the universal gravitational constant,1.069304x10^-9ft.^3/lb.sec.^2,
M = the mass of the earth, 1.31672x10^25 lb., and r = the radius of the earth in feet, 5280(3963.4) = 20,926,752 feet, resulting in a mean g = g = 32.519ft./sec.6@.
As written, these expressions apply to falling bodies.
The equations that apply to rising bodies are
....Vf = Vo - gt (the term “g” for acceleration due to gravity is assumed constant on, or near, the surface of the Earth)
....h = Vo(t)-g(t^2)/2
....Vf^2 = Vo^2-2gh
All of the above ignores surface friction.
The height reached then becomes
h = Vo(t)-32.519(t^2)/2 where
t = (Vo - Vf)32.519 or
h = (Vo^2 - Vf^2)65.04
To determine how far above the surface the projectile will go, we need to find the maximum height it reaches.
Step 1: Find the initial kinetic energy of the projectile.
The initial kinetic energy is given by the formula KE = 1/2 * m * v0^2, where m is the mass of the projectile and v0 is the initial velocity.
Since the mass of the projectile is not mentioned, we can assume it to be m.
Step 2: Find the potential energy at the maximum height.
When the projectile reaches its maximum height, all its initial kinetic energy is converted into potential energy.
The potential energy at any height h above the surface of the Earth is given by PE = m * g * h, where g is the acceleration due to gravity.
Step 3: Equate the initial kinetic energy to the potential energy at the maximum height.
Setting the initial kinetic energy equal to the potential energy, we have:
1/2 * m * v0^2 = m * g * h
Step 4: Solve for h.
Dividing both sides of the equation by m and rearranging, we get:
h = (v0^2) / (2 * g)
Step 5: Substitute the known variables.
Using the given variables and constants, the expression for h becomes:
h = (v0^2) / (2 * G * ME / rE)
Therefore, the height above the surface of the Earth that the projectile will go is given by the equation:
h = (v0^2) / (2 * G * ME / rE)
To determine how far above the Earth's surface the projectile will go, we can use the principle of conservation of energy. The initial kinetic energy of the projectile is given by 1/2 * m * v0^2, where m is the mass of the projectile and v0 is the initial speed.
As the projectile goes higher, its potential energy increases while its kinetic energy decreases. At the highest point of its trajectory, all of its initial kinetic energy is converted to potential energy. Therefore, we can equate the two energies:
1/2 * m * v0^2 = m * g * h
Where g is the acceleration due to gravity and h is the height above the Earth's surface.
To solve for h, we need to express the gravitational acceleration in terms of the variables mentioned. The magnitude of the gravitational acceleration is given by g = G * ME / r^2, where G is the gravitational constant, ME is the mass of the Earth, and r is the distance between the projectile and the center of the Earth, which is equal to the radius of the Earth plus h.
Substituting this expression for g in the equation above:
1/2 * m * v0^2 = m * (G * ME / (rE + h)^2) * h
Cancelling out the mass of the projectile and rearranging the equation to solve for h, we have:
v0^2 / (2 * G * ME / (rE + h)^2) = h
To simplify further, we can multiply both sides by (rE + h)^2 and divide by 2:
v0^2 * (rE + h)^2 / (2 * G * ME) = h
This equation gives us the height above the Earth's surface that the projectile will reach. Plug in the given values for v0, rE, ME, and G to find the numerical value of h.