Zero, a hypothetical planet, has a mass of 3.0*10^23 kg, a radius of 3.0*10^6 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface.
(a) If the probe is launched with an initial kinetic energy of 5.0*10^7 J, what will be its kinetic energy when it is 4.0*10^6 m from the center of Zero?
(b)If the probe is to achieve a maximum distance of 8.0*10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
***for letter A, i got 2.335e8 by E = KE - GMm/R. What am i doing wrong?
thanks
SORRY FOR DOUBLE POST
On A, You have to remember that the change in PE is GMm/R - GMm/radius
Where R is 4.0*10^6 m and radius is the intial radius, or 3.0*10^6.
E= KEinitial -GMm/R + GMm/radius
On A, You have to remember that the change in PE is GMm/R - GMm/radius
Where R is 4.0*10^6 m and radius is the intial radius, or 3.0*10^6.
E= KEinitial -GMm/R + GMm/radius
To calculate the change in kinetic energy of the space probe when it is 4.0*10^6 m from the center of Zero, we can use the conservation of mechanical energy.
The initial kinetic energy of the space probe is given as 5.0*10^7 J. This is the sum of the initial gravitational potential energy and the initial kinetic energy:
E_initial = KE_initial + PE_initial
The formula for gravitational potential energy is given by PE = -GMm/r, where G is the gravitational constant, M is the mass of the planet Zero, m is the mass of the space probe, and r is the distance from the center of Zero.
At the surface of Zero (radius = 3.0*10^6 m), the potential energy is:
PE_initial = -GMm/radius
When the space probe is 4.0*10^6 m from the center of Zero (R = 4.0*10^6 m), the potential energy is:
PE_final = -GMm/R
The change in potential energy is then:
delta_PE = PE_final - PE_initial = (-GMm/R) - (-GMm/radius)
Substituting the values given, we have:
delta_PE = (-GMm/4.0*10^6) - (-GMm/3.0*10^6)
Simplifying further, we have:
delta_PE = GMm(1/R - 1/radius)
Using the equation E_final = KE_final + PE_final, we can express the change in kinetic energy as:
delta_KE = E_final - E_initial = KE_final - KE_initial = delta_PE
So, the kinetic energy when the space probe is 4.0*10^6 m from the center of Zero is:
KE_final = delta_PE + KE_initial = delta_KE + KE_initial
Substituting the values given, we have:
KE_final = delta_KE + 5.0*10^7 J = GMm(1/R - 1/radius) + 5.0*10^7 J