Consider a cannon, shooting projectiles with speed vi, straight up in the air. W

Question

Consider a cannon, shooting projectiles with speed vi, straight up in the air. We assume that the Earth is spherical, and has radius Re. The force of gravity is given by
Fg = GMm/r^2. (1)
where M is the mass of the Earth (look it up), m = 5.00 kg is the mass of the projectile, and G is Newton’s constant (look it up).
a) Explicitly compute the work done by gravity on the projectile, as it flies from Re with speed vi, to some Rf, with speed vf , and relate it to the kineticenergy.
b) We define the escape velocity to be the value that vi should have, such that vf 0 as Rf . What is the escape velocity for the Earth? For the Sun?
c) The Schwarzschild radius of a gravitating body of mass M is the value Rs, such that had the launch radius been Rs, the escape velocity would have been equal to the speed of light. What is Rs for the Earth? For the Sun? Gravitating bodies that are smaller than their Schwarzschild radius are called Black Holes".
Consider a cannon, shooting projectiles with speed vi, straight up in the air. We assume that the Earth is spherical, and has radius Re. The force of gravity is given by
Fg = GMm/r^2. (1)
where M is the mass of the Earth (look it up), m = 5.00 kg is the mass of the projectile, and G is Newton’s constant (look it up).
a) Explicitly compute the work done by gravity on the projectile, as it flies from Re with speed vi, to some Rf, with speed vf , and relate it to the kineticenergy.
b) We define the escape velocity to be the value that vi should have, such that vf 0 as Rf . What is the escape velocity for the Earth? For the Sun?
c) The Schwarzschild radius of a gravitating body of mass M is the value Rs, such that had the launch radius been Rs, the escape velocity would have been equal to the speed of light. What is Rs for the Earth? For the Sun? Gravitating bodies that are smaller than their Schwarzschild radius are called Black Holes".
Consider a cannon, shooting projectiles with speed vi, straight up in the air. We assume that the Earth is spherical, and has radius Re. The force of gravity is given by
Fg = GMm/r^2. (1)
where M is the mass of the Earth (look it up), m = 5.00 kg is the mass of the projectile, and G is Newton’s constant (look it up).
a) Explicitly compute the work done by gravity on the projectile, as it flies from Re with speed vi, to some Rf, with speed vf , and relate it to the kineticenergy.
b) We define the escape velocity to be the value that vi should have, such that vf 0 as Rf . What is the escape velocity for the Earth? For the Sun?
c) The Schwarzschild radius of a gravitating body of mass M is the value Rs, such that had the launch radius been Rs, the escape velocity would have been equal to the speed of light. What is Rs for the Earth? For the Sun? Gravitating bodies that are smaller than their Schwarzschild radius are called Black Holes". Consider a cannon, shooting projectiles with speed vi, straight up in the air. We assume that the Earth is spherical, and has radius Re. The force of gravity is given by
Fg = GMm/r^2. (1)
where M is the mass of the Earth (look it up), m = 5.00 kg is the mass of the projectile, and G is Newton’s constant (look it up).
a) Explicitly compute the work done by gravity on the projectile, as it flies from Re with speed vi, to some Rf, with speed vf , and relate it to the kineticenergy.
b) We define the escape velocity to be the value that vi should have, such that vf 0 as Rf . What is the escape velocity for the Earth? For the Sun?
c) The Schwarzschild radius of a gravitating body of mass M is the value Rs, such that had the launch radius been Rs, the escape velocity would have been equal to the speed of light. What is Rs for the Earth? For the Sun? Gravitating bodies that are smaller than their Schwarzschild radius are called Black Holes".

Explanation / Answer

a) work done by gravity = force x dsiaplcement = Fg ( Rf- Ri)= -GmM/ Re^2 ( Rf-ri)

Initial Energy = -GmM/ Re + 1.2 mvi^2

Final energy = - GmM/Rf + 1/2 mvf^2

work done = chnage in energy

-GmM/ Re^2 ( Rf-ri) =  GmM/Rf + 1/2 mvf^2 +  GmM/ Re- 1.2 mvi^2

g = GM/ Re^2= 9.8

- 9.8 ( 5) ( Rf - Ri) = GmM ( 1/Rf- 1/Re) + 0.5 ( 5) ( Vf^2 - v^2)

-48 (Rf-Re) = GmM ( 1/Rf- 1/Re) + 0.5 ( 5) ( Vf^2 - v^2) - work done by gravity

b) V escape = sqroot ( 2GM/ R)

For earth = sqroot ( 2x 6,674 x 10^ -11 x 6 x 10^24 / 6371 x 10^3) = sqroot ( 0.01257 x 10^ 10) = 0.1121 x 10^5 = 11.2 km /s

For su n= sqroot ( 2x 6.674 x 10^ -11 x 2 x 10^30 /  695,700 x 10^3) = sqroot (3.84 x 10^11) = 6.1967 x 10^5 = 619.67 km/s

c) Schwarzschild radius = 2GM/ c^2

for earth = 2x 6,674 x 10^ -11 x 6 x 10^24/ 9 x 10^16 = 8.898 x 10^ -3 m

for sun= 2x 6.674 x 10^ -11 x 2 x 10^30 / 9 x 10^ 16 = 2.966 x 10^3 m

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