A spaceship of mass mm circles a planet of mass M in an orbit of radius R. How much energy is required to transfer the spaceship to a circular orbit of radius 3R?

ΔE = GmM/3R

Explanation:

The absolute potential energy of an object in a planet's field is given as:

E = -GmM/2r

where,

E = Potential Energy

G = Universal Gravitational Constant

m = mass of spaceship

M = Mass of Planet

r = distance from surface of planet

Therefore, for initial state:

E = E₁ and r = R

E₁ = - GmM/2R

and for final state:

E = E₂ and r = 3R

E₂ = - GmM/6R

So, the required energy will be:

ΔE = E₂ - E₁ = - GmM/6R + GmM/2R

ΔE = GmM(- 1/6R + 1/2R)

ΔE = GmM/3R

the answer is a. the yacht traveled from the dock north to the 200-meter marker in the bay in less than 5 minutes, giving the passengers several more hours to fish.

im doing peak rn so sry if its wrong

the minimum constant speed at which she can catch the train is 2.7 m/s

given:

acceleration = 0.6

time = 4.5 s

to find:

velocity = ?

formula used:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

solution:

from 1st kinematic equation we get,

v = u + at

u = initial speed of train = 0 m/s

v = final speed of the train

a = acceleration of the train

t = time

v = 0 + (0.6 × 4.5)

v = 2.7 m/s

the minimum constant speed at which she can catch the train is 2.7 m/s