Sadly you have become trapped inside the gravity well of a star of mass M = 6.435×10^30kg. At a distance of r0 = 7.543×10^10m from the center of the star you find yourself moving directly towards it at 5.876×10^4m/s (relative to the star). How fast are you moving relative to the star when you inexorably find yourself a distance of rf=1.460×10^10m from its center? (You have no form of mass/energy that you can throw/eject/propel/fire/burn towards the star to slow or prevent your gravitational free-fall towards the star.)
Sorry about not replying last time! I figured this one out though.
The solution can be found by:
vf=√[(v0^2)+(2GM/rf)-(2GM/r0)]
Thanks! Got it.
Did you get #1 and #3?
To determine the speed at distance rf from the center of the star, we can use the conservation of mechanical energy in a gravitational field.
1. Find the initial total mechanical energy (E0) at distance r0 from the center of the star:
The total mechanical energy is the sum of kinetic energy (KE0) and gravitational potential energy (PE0).
KE0 = 1/2 * m * v0^2, where m is your mass and v0 is the initial velocity relative to the star.
PE0 = -G * M * m / r0, where G is the gravitational constant, M is the mass of the star, and r0 is the initial distance from the center of the star.
E0 = KE0 + PE0.
2. Find the final potential energy (PEf) at distance rf from the center of the star:
PEf = -G * M * m / rf.
3. Apply the conservation of mechanical energy:
E0 = KEf + PEf, where KEf is the final kinetic energy and E0 is the initial total mechanical energy.
Since we have no means to change our kinetic energy (no propulsion mechanism), the value of KEf should be the same as KE0.
4. Solve for the final velocity (vf) relative to the star:
vf = √(2 * (E0 - PEf) / m), where E0 is the initial total mechanical energy and PEf is the final potential energy.
Now we can plug in the given values and calculate the result:
Given:
- M = 6.435×10^30 kg (mass of the star)
- r0 = 7.543×10^10 m (initial distance from the center of the star)
- v0 = 5.876×10^4 m/s (initial velocity relative to the star)
- rf = 1.460×10^10 m (final distance from the center of the star)
Let's calculate the final velocity (vf):
1. Calculate the initial energy (E0):
KE0 = 1/2 * m * v0^2
KE0 = 0.5 * m * (5.876×10^4)^2
PE0 = -G * M * m / r0
PE0 = -6.67430×10^-11 * 6.435×10^30 * m / (7.543×10^10)
E0 = KE0 + PE0
2. Calculate the final potential energy (PEf):
PEf = -G * M * m / rf
PEf = -6.67430×10^-11 * 6.435×10^30 * m / (1.460×10^10)
3. Calculate the final velocity (vf):
vf = √(2 * (E0 - PEf) / m)
vf = √(2 * (E0 - PEf) / m)
Substitute the values of E0, PEf, and m, then calculate vf.
By following this process, you can determine the final velocity relative to the star when you reach a distance of rf from its center.