How high does it go?
I keep getting 885km, but supposedly its wrong…
Thanks
1 Answer

first convert this speed to m/s; 1.5×10^4km/hr= 4166.7m/s
the speeds involved in this problem will cause the projectile to rise high enough that we cannot regard gravity as a constant, so we can’t use g=9.8m/s/s since the projectile will rise high enough such that the value of g will vary significantly during the projectile’s trajectory
we can use the formulation of conservation of energy in which we define PE as GMm/r where G is the newtonian grav cst, M is the mass of the earth, m is the mass of the projectile and r is the distance of the projectile from the center of the earth
on the surface of the earth, the projectile has KE of 1/2 mv^2 and PE of GMm/Re where Re is the radius of the earth
at its apex, v=0 and PE =GMm/r where r is the distance of the highest point from the center of the earth.
equating energies, we have
1/2 mv^2 – GMm/Re = – GMm/r or
v^2 = 2GM(1/Re1/r)
using v=4.17 x10^3m/s
G=6.67×10^(11)in MKS units
M=6×10^24kg
we get (1/Re1/r) = 2.17×10^(8)
Re=6.4×10^6 m so 1/r = 1.34×10^(7) so r=7.43×10^6m
this is the distance from the center of the earth, or (7.436.4)x10^6=1.03×10^6 m
above the surface of the earth; this is 1031 km above the surface of the earth