Answer:
(a) h = 1.27 Re
(b) h = 1.86 Re
Explanation:
Let M is the mass of earth and Re is the radius of earth.
initial velocity of projection, v = 0.462 ve
where, ve is the escape velocity of an object on earth surface.
(a)
The value of escape velocity is
![v_{e}=\sqrt{\frac{2GM}{R_{e}}}](https://tex.z-dn.net/?f=v_%7Be%7D%3D%5Csqrt%7B%5Cfrac%7B2GM%7D%7BR_%7Be%7D%7D%7D)
So,
.... (1)
By using conservation of energy
(Kinetic energy + potential energy ) at the surface of earth = Potential energy at the height h.
where, h is the maximum height upto which the projectile reach
K.E at surface P.E at surface = P.E at the top
![\frac{1}{2}mv^{2}-\frac{GMm}{R_{e}}=-\frac{GMm}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dmv%5E%7B2%7D-%5Cfrac%7BGMm%7D%7BR_%7Be%7D%7D%3D-%5Cfrac%7BGMm%7D%7Bh%7D)
By equation (1), substituting the value of v
![\frac{1}{2}\times 0.462^{2}\times \frac{2GM}{R_{e}}-\frac{GM}{R_{e}}=-\frac{GM}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%200.462%5E%7B2%7D%5Ctimes%20%5Cfrac%7B2GM%7D%7BR_%7Be%7D%7D-%5Cfrac%7BGM%7D%7BR_%7Be%7D%7D%3D-%5Cfrac%7BGM%7D%7Bh%7D)
![\frac{1}{2}\times 0.462^{2}\times \frac{2}{R_{e}}-\frac{1}{R_{e}}=-\frac{1}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%200.462%5E%7B2%7D%5Ctimes%20%5Cfrac%7B2%7D%7BR_%7Be%7D%7D-%5Cfrac%7B1%7D%7BR_%7Be%7D%7D%3D-%5Cfrac%7B1%7D%7Bh%7D)
h = 1.27 Re
(b)
initial kinetic energy = 0.462 times the kinetic energy required to escape
![\frac{1}{2}mv^{2}=0.462\times \frac{1}{2}mv_{e}^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dmv%5E%7B2%7D%3D0.462%5Ctimes%20%5Cfrac%7B1%7D%7B2%7Dmv_%7Be%7D%5E%7B2%7D)
![\frac{1}{2}mv^{2}=0.462\times \frac{1}{2}m\times \frac{2GM}{R_{e}}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dmv%5E%7B2%7D%3D0.462%5Ctimes%20%5Cfrac%7B1%7D%7B2%7Dm%5Ctimes%20%5Cfrac%7B2GM%7D%7BR_%7Be%7D%7D)
So, again by using the conservation of energy
Kinetic energy at the surface + Potential energy at the surface = Potential energy at the top
![0.462\times \frac{1}{2}m\times \frac{2GM}{R_{e}}-\frac{GMm}{R_{e}}=-\frac{GMm}{h}](https://tex.z-dn.net/?f=0.462%5Ctimes%20%5Cfrac%7B1%7D%7B2%7Dm%5Ctimes%20%5Cfrac%7B2GM%7D%7BR_%7Be%7D%7D-%5Cfrac%7BGMm%7D%7BR_%7Be%7D%7D%3D-%5Cfrac%7BGMm%7D%7Bh%7D)
![0.462\times \frac{1}{R_{e}}-\frac{1}{R_{e}}=-\frac{1}{h}](https://tex.z-dn.net/?f=0.462%5Ctimes%20%5Cfrac%7B1%7D%7BR_%7Be%7D%7D-%5Cfrac%7B1%7D%7BR_%7Be%7D%7D%3D-%5Cfrac%7B1%7D%7Bh%7D)
h = 1.86 Re