A projectile is launched vertically from the surface of the Moon with an initial speed of 1130 m/s. At what altitude is the projectile's speed three-fourths its initial value?
change in PE=change in KE
PE=1/2 mass(1130^2-(3*1130/4)^2 )
Now PE change is Mass*GMm/(altitude+radiusmoon)
so you have to look up the mass of the moon
Notice the mass divides out.
To solve this problem, we can use the principle of conservation of mechanical energy. The total mechanical energy of a projectile on the Moon's surface is given by the equation:
E = KE + PE
Where:
E = Total mechanical energy
KE = Kinetic energy
PE = Potential energy
When the projectile is launched vertically from the surface of the Moon, its initial kinetic energy KE_0 is given by:
KE_0 = 1/2 * m * v_0^2
Where:
m = Mass of the projectile
v_0 = Initial speed of the projectile
Initially, the potential energy PE_0 is zero since the projectile is on the surface of the Moon.
As the projectile reaches a certain altitude, let's call it h, its potential energy becomes non-zero. At this point, we can equate the initial kinetic energy with the sum of final kinetic energy and potential energy:
KE_0 = KE_f + PE_f
Since the projectile is still in motion, its final kinetic energy KE_f is given by:
KE_f = 1/2 * m * v_f^2
Where:
v_f = Final speed of the projectile
Also, at this point, the potential energy PE_f becomes non-zero and is given by:
PE_f = m * g * h
Where:
g = Acceleration due to gravity on the Moon
h = Altitude at which the projectile's speed is three-fourths its initial value
We can now substitute the equations for final kinetic energy and potential energy into the conservation of mechanical energy equation:
KE_0 = KE_f + PE_f
1/2 * m * v_0^2 = 1/2 * m * v_f^2 + m * g * h
Since the projectile is launched vertically, the final speed v_f is given by:
v_f = v_0 - g * t
Where:
t = Time taken by the projectile to reach altitude h
We want to find the altitude h at which the projectile's speed is three-fourths its initial value:
v_f = 3/4 * v_0
Substituting this condition into the equation for v_f:
3/4 * v_0 = v_0 - g * t
Simplifying this equation, we can solve for the time t:
t = v_0 / (4 * g)
Finally, we substitute the value of t into the equation for h:
h = (v_0^2 / (2 * g)) - (v_f * t)
Substituting the values given in the problem:
v_0 = 1130 m/s
g = Acceleration due to gravity on the Moon
We can now calculate the altitude h.