Answer:
The value is [tex]v = 2.3359 *10^{4} \ m/s[/tex]
Explanation:
From the question we are told that
The initial speed is [tex]u = 2.05 *10^{4} \ m/s[/tex]
Generally the total energy possessed by the space probe when on earth is mathematically represented as
[tex]T__{E}} = KE__{i}} + KE__{e}}[/tex]
Here [tex]KE_i[/tex] is the kinetic energy of the space probe due to its initial speed which is mathematically represented as
[tex]KE_i = \frac{1}{2} * m * u^2[/tex]
=> [tex]KE_i = \frac{1}{2} * m * (2.05 *10^{4})^2[/tex]
=> [tex]KE_i = 2.101 *10^{8} \ \ m \ \ J[/tex]
And [tex]KE_e[/tex] is the kinetic energy that the space probe requires to escape the Earth's gravitational pull , this is mathematically represented as
[tex]KE_e = \frac{1}{2} * m * v_e^2[/tex]
Here [tex]v_e[/tex] is the escape velocity from earth which has a value [tex]v_e = 11.2 *10^{3} \ m/s[/tex]
=> [tex]KE_e = \frac{1}{2} * m * (11.3 *10^{3})^2[/tex]
=> [tex]KE_e = 6.272 *10^{7} \ \ m \ \ J[/tex]
Generally given that at a position that is very far from the earth that the is Zero, the kinetic energy at that position is mathematically represented as
[tex]KE_p = \frac{1}{2} * m * v^2[/tex]
Generally from the law energy conservation we have that
[tex]T__{E}} = KE_p[/tex]
So
[tex]2.101 *10^{8} m + 6.272 *10^{7} m = \frac{1}{2} * m * v^2[/tex]
=> [tex]5.4564 *10^{8} = v^2[/tex]
=> [tex]v = \sqrt{5.4564 *10^{8}}[/tex]
=> [tex]v = 2.3359 *10^{4} \ m/s[/tex]
