Grade 12 Gravitational Force Fields Questions.

[TheSerpent]

1.Nasa fires a 1 tonne space probe, from a stable orbit of altitude 13,620km using a giant rail gun. The rail gun provides the probe with 9.97165*10^10 J of additional kinetic energy. Determine if the probe has enough velocity to escape the Earth's gravity well. The Earth's mass and radius are 5.98*10^24kg and 6380km respectively.

 

[plancking]

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)
E_kinetic=0.5mv²

m (rocket)=1000 kg
r (orbit)=13620 km
E_kinetic=9.97165*10¹⁰ J
m (earth)=5.98*10²⁴ kg
r (earth)=6380 km

First, find the velocity that the rocket has...
E_kinetic=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/r_orbit+r_earth)
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]

First determine the initial velocity of the space probe in its original orbit (prior to receiving a boost from the rail gun). Remember that you're given its altitude, not its orbit radius.

From that velocity determine what the escape velocity would be; escape velocity has a particular relationship to the circular orbit velocity. Then determine what the new velocity would be given the rail gun boost of energy. Compare to the escape velocity.

 

[gneill]

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)
E_kinetic=0.5mv²

m (rocket)=1000 kg
r (orbit)=13620 km <--- That's its altitude, not its orbital radius
E_kinetic=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...
E_kinetic=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/r_orbit+r_earth)
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.

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]

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.

Thanks for the correction!

 

[TheSerpent]

Thanks for the correction!

So what's going to be changed in the calculation process (how do you find the new velocity?)

 

[gneill]

So what's going to be changed in the calculation process (how do you find the new velocity?)

The rail gun adds kinetic energy. Velocity depends upon kinetic energy.

 

[TheSerpent]

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.

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]

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

Okay! Good to here!

How would you include the energy boost from the rail gun into the answer, that is what I am not sure of.

Kinetic energy adds. Add the boost energy to the energy k that you found above. Find the new velocity from that.

 

[TheSerpent]

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.

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]

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.

Yup. That looks good.

 

[TheSerpent]

Yup. That looks good.

Thank you for your help! Can you also help us in our other post you commented on? We posted an idea but unsure!

 

[gneill]

Thank you for your help! Can you also help us in our other post you commented on? We posted an idea but unsure!

You're welcome.

I see that you have two threads, this one and one other. I've made a suggestion for you to work on in that other thread.