plancking said:Hi TheSerpent,
The formula for escape velocity is v=√(2GM/r) where v= velocity, G=universal gravitational constant (G=6.67*10-11 Nm2kg-2), M is the mass of the body, and r is the separation distance.
v=√(2GM/r)
Ekinetic=0.5mv2
m (rocket)=1000 kg
r (orbit)=13620 km <--- That's its altitude, not its orbital radius
Ekinetic=9.97165*1010 J <--- That's the energy added, not the total
m (earth)=5.98*1024 kg
r (earth)=6380 km
First, find the velocity that the rocket has...
Ekinetic=0.5mv2
9.97165*1010 J = 0.5(1000 kg)v2
v=√((2*(9.97165*1010 J))/1000 kg)
v=1.412*104 m/s
Then, find the velocity that the rocket would need to escape...
v=√(2GM/r)
v=√(2GM/rorbit+rearth)
v=√((2*(6.67*10-11 Nm2kg-2)*(5.98*1024 kg))/(13620 km + 6380 km))
v=1.997*105 m/s
Compare the two -- the rocket will not have sufficient velocity to escape.
Good luck! I hope this helps.
gneill said:The rocket begins in a circular orbit at the given altitude. Therefore it will have an initial velocity and kinetic energy appropriate for a circular orbit at that location. Next additional energy is supplied via a rail gun boost. The probe will gain that energy as KE and thus have a new velocity. Compare to the escape velocity for that orbital radius.
plancking said:Thanks for the correction!
TheSerpent said:So what's going to be changed in the calculation process (how do you find the new velocity?)
gneill said:The rocket begins in a circular orbit at the given altitude. Therefore it will have an initial velocity and kinetic energy appropriate for a circular orbit at that location. Next additional energy is supplied via a rail gun boost. The probe will gain that energy as KE and thus have a new velocity. Compare to the escape velocity for that orbital radius.
Okay! Good to here!TheSerpent said:Find orbital velocity:
v = √(GM/r)
v = √((6.67*10^-11)(5.98*10^24)/(13620000 + 6380000)
v = 4.466 * 10^3 m/s
Find orbital kinetic energy:
K = mv^2/2
k = (1000)(4.466*10^3)^2/2
K = 9972578000 = 9.973 * 10^9 J
Find escape velocity:
v esc = √2 v
v esc = √2 (4.466 * 10^3)
v esc = 6.316 * 10^3 m/s
How would you include the energy boost from the rail gun into the answer, that is what I am not sure of.
gneill said:Okay! Good to here!
Kinetic energy adds. Add the boost energy to the energy k that you found above. Find the new velocity from that.
TheSerpent said:Find new velocity:
K = mv^2 / 2
(9.973 * 10^9)+(9.97165 * 10^10) = (1000)v^2 / 2
v = √(2)(1.096895*10^11)/1000
v = 14811.44827 = 1.481 * 10^4 m/s
This is the actual velocity so therefore the rocket will have sufficient velocity to escape.
gneill said:Yup. That looks good.
TheSerpent said:Thank you for your help! Can you also help us in our other post you commented on? We posted an idea but unsure!
Gravitational force is the attractive force between two objects with mass. It works by pulling objects towards each other in proportion to their masses and the distance between them.
Newton's Law of Universal Gravitation states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In other words, the greater the masses of the objects and the closer they are, the stronger the gravitational force between them.
Gravitational force is a universal force that acts on all objects with mass, regardless of their size or composition. It is also a conservative force, meaning it does not dissipate energy and can be converted into other forms of energy.
Scientists can measure gravitational force using a device called a gravimeter, which measures the acceleration due to gravity. They can also use Newton's Law of Universal Gravitation to calculate the force between two objects based on their masses and distance.
Gravitational force cannot be shielded or canceled out, as it is a fundamental force of nature. However, its effects can be counteracted by other forces, such as electromagnetic force, which is why objects can appear to be weightless in space where there is minimal gravitational force.