Trajectory physics question (kicking a ball)?

Question

I could really use some help with this problem that I can't seem to solve after spending several hours on it.

A soccer player kicks a ball from level ground so that it will reach its maximum range, which is Rmax. How high is the ball when its horizontal distance from the starting point is 1/2 Rmax? How high is the ball when its horizontal distance from the starting point is 3/4 Rmax? (Ignore wind resistance).

This is not a mass question b/c we haven't covered that yet. We're only on vectors and trajectories.

Answer 1

h= (V*sine x)^2/(2*g) at 0.5 Rmax.-----(1)
Also h=0.5*g*t1^2-----(2)
t2=t1 ----(3), where V is initial velocity at angle x, h=maximum height, and t1 and t2 are rising and falling times.
Vh=V*cos x, where Vh is horizontal velocity component.
Rmax=Vh*(t1+t2)
Rmax=2*t1*Vh
Rmax=2*V*cos x*t1
t1= 0.5*Rmax/(V*cos x) ---(4)
Inserting (4) in (2), we get:
h= 0.5*g*[0.5*Rmax/(V*cos x)]^2 (at 1/2Rmax)
h=0.5*g*[0.5*3/2*Rmax/(V*cos x)]^2 ( at 3/4Rmax)

Answer 2

The equation F = Ma (10-1) is for Mass only, (under micro-Gravity conditions understood). On Earth or where there is a defined weight, the equation should be F = Wa (10-2)). Current textbooks do not explain nor distinguish the difference, mostly because that standard weight is called Mass.
Say a human kicks a 15 pound Earth weight Mass. It would break that person's toes. That same Mass taken aboard our space shuttle, in orbit, could be kicked with impunity. Taken to the Moon, it could still be kicked with no problems. (Earth weight about 2.5#) Taken to Saturn or Jupiter it could not be kicked at all or its Inertia, to that person kicking, would be infinite. Likewise and just as obvious, that 15# weight could be made into a lead disk and still be kicked on the Earth (hit it dead center) and not break ones toes, but might bruise them a bit, depending on, in all cases, the TIME RATE OF APPLICATION of the kick so given, i.e., speed of foot (toes) at point of contact with the Mass in question.
TV pictures of one of the astronauts, accidentally or on purpose, kicking a rather large boulder, about 12” across, on the Moon’s surface, shows that it just bounced away, as if it was a soccer ball. At the risk of being repetitious, this likewise disapproves Mach’s Postulate on that Inertia is a resultant of Forces from all the Mass in space. Simply, the position of the Mass would result in a zero value for Inertia regardless of where it was located in space. See CHAPTER 15 where this relates to Push Gravity due to some outside source.

Answer 3

the ball will attain rmax on condition that kiced at 45degree. if ball is kicked with speed v then we use vcos for horizontal and vsin for vertical . now a million/2 r = vcos t = v sqrt 2 t or t = a million/2 r/v sqrt 2. now to locate the vertical authentic s = a million/2 g t^2 use the t calculated in the previous to get the answer. do similar for 3/4r.