Standing on Eris, a dwarf planet of raidus of 1200km, drop a rock from height of 0.30m. The rock takes 0.87s to reach the surface.

a) Calculate the value of g on Eris

b) Calculate the mass of Eris

c) Show the ratio of your altitude above any planet-to-the ratio of the same planet for when your weight is one-half your weight on the surface always equals the same expression: r:rp = sqr(2):1

(1/2) m v^2 = m g h

so v= sqrt(2 g h) at ground
v average during fall = (1/2) sqrt(2 g h)
0.3meters = v average * t = (1/2) sqrt(2 g * 0.3) * 0.87
0.6 / 0.87 = sqrt (0.6 g )
solve for g
G m M / R^2 = m g
G = 6.7*10^-11
R = 1.2 * 10^6
solve for M

Let's calculate the answers step by step:

a) Calculate the value of g on Eris:
The acceleration due to gravity can be calculated using the formula:
g = (2 * h) / (t^2)

Where:
g is the acceleration due to gravity,
h is the height (0.30m in this case),
t is the time taken to reach the surface (0.87s in this case).

Plugging in the values, we get:
g = (2 * 0.30) / (0.87^2)
g = 0.689 m/s^2

Therefore, the value of g on Eris is 0.689 m/s^2.

b) Calculate the mass of Eris:
The mass of a planet can be calculated using the formula:
m = (g * r^2) / G

Where:
m is the mass of the planet,
g is the acceleration due to gravity (0.689 m/s^2),
r is the radius of the planet (1200 km or 1,200,000 meters),
G is the universal gravitational constant (6.67430 × 10^-11 N*(m/kg)^2).

Plugging in the values and converting the radius to meters, we get:
m = (0.689 * (1,200,000)^2) / (6.67430 × 10^-11)
m = 1.7198 × 10^23 kg

Therefore, the mass of Eris is approximately 1.7198 × 10^23 kg.

c) Show the ratio of your altitude above any planet-to-the ratio of the same planet for when your weight is one-half your weight on the surface always equals the same expression: r:rp = sqr(2):1
In this case, the ratio is described as:
(r / rp) = (√2) / 1

This means that the ratio of your altitude above any planet to the ratio of the same planet, where your weight is one-half your weight on the surface, is always equal to √2:1.

a) To calculate the value of g on Eris, we can use the formula for acceleration due to gravity: g = (2 * h) / t^2, where h is the drop height and t is the time taken.

Given:
h = 0.30 m
t = 0.87 s

Substituting the values into the formula:
g = (2 * 0.30) / (0.87)^2
g = 0.693 m/s^2

Therefore, the value of g on Eris is approximately 0.693 m/s^2.

b) To calculate the mass of Eris, we can use Newton's law of universal gravitation: F = (G * m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.

Given:
Raidus of Eris (r) = 1200 km = 1,200,000 meters
Gravitational force (F) = weight of the rock = m * g

We already calculated g as 0.693 m/s^2. Now, we need to convert the weight from m/s^2 to kg/s^2, so we divide it by 9.8 (acceleration due to gravity on Earth).

Weight (W) = m * g = m * 0.693 / 9.8

Now, substituting the values into the formula and solving for m (mass of Eris):
W = (G * m1 * m2) / r^2
m1 = mass of the rock = Weight / g

Simplifying:
m1 = Weight / g = W / 0.693 / 9.8

Substituting the known values of W and r:
m2 = (F * r^2) / (G * m1)
= (m * g * r^2) / (G * m1)
= (m * g * r^2) / (G * (W / g))

We can cancel out the factors of mass (m):
m2 = (g * r^2) / (G * (1 / g))
= (g * r^2 * g) / (G)

Now, substitute the known values:
m2 = (0.693 * (1,200,000)^2 * 0.693) / G

The value of G is approximately 6.67430 x 10^-11 N m^2/kg^2 (gravitational constant).

Substituting this value:
m2 = (0.693 * (1,200,000)^2 * 0.693) / (6.67430 x 10^-11)
m2 ≈ 1.648 x 10^23 kg

Therefore, the mass of Eris is approximately 1.648 x 10^23 kg.

c) The expression given is r : rp = √2 : 1. In this context, r represents the altitude above any planet, and rp represents the radius of the same planet.

The ratio of weight (W) on the surface to the weight (W') at an altitude above any planet can be related to the ratio of radius to (radius + altitude).

Mathematically, the ratio of weights is given as:

W / W' = rp^2 / (r + rp)^2

Using the given expression, r : rp = √2 : 1, we can substitute this value into the formula:

W / W' = ( (√2)^2 * rp^2 ) / ( (√2 * rp + rp)^2 )
W / W' = 2 * rp^2 / ( (rp * (√2 + 1))^2 )

Simplifying further:
W / W' = 2 * rp^2 / ( (rp * (1 + √2))^2 )
W / W' = 2 * rp^2 / ( rp^2 * (1 + √2)^2 )

We can cancel out the rp^2 terms:
W / W' = 2 / (1 + √2)^2
W / W' = 2 / (1 + 2√2 + 2)

Simplifying:
W / W' ≈ 2 / (3 + 2√2)

Therefore, the ratio r : rp = √2 : 1 will always equal the expression 2 / (3 + 2√2).