Answer:
H(max) = (v²/2g)
Explanation:
The maximum height the ball will climb will be when there is no friction at all on the surface of the hill.
Normally, the conservation of kinetic energy (specifically, the work-energy theorem) states that, the change in kinetic energy of a body between two points is equal to the work done in moving the body between the two points.
With no frictional force to do work, all of the initial kinetic emergy is used to climb to the maximum height.
ΔK.E = W
ΔK.E = (final kinetic energy) - (initial kinetic energy)
Final kinetic energy = 0 J, (since the body comes to rest at the height reached)
Initial kinetic energy = (1/2)(m)(v²)
Workdone in moving the body up to the height is done by gravity
W = - mgH
ΔK.E = W
0 - (1/2)(m)(v²) = - mgH
mgH = mv²/2
gH = v²/2
H = v²/2g.
Missing question:
"Determine (a) the astronaut’s orbital speed v and (b) the period of the orbit"
Solution
part a) The center of the orbit of the third astronaut is located at the center of the moon. This means that the radius of the orbit is the sum of the Moon's radius r0 and the altitude (

) of the orbit:

This is a circular motion, where the centripetal acceleration is equal to the gravitational acceleration g at this altitude. The problem says that at this altitude,

. So we can write

where

is the centripetal acceleration and v is the speed of the astronaut. Re-arranging it we can find v:

part b) The orbit has a circumference of

, and the astronaut is covering it at a speed equal to v. Therefore, the period of the orbit is

So, the period of the orbit is 2.45 hours.
Answer:
centripetal force is a net force that acts on an object to keep it moving along a circular path and elastic force acts to return a spring to its natural length