How to solve final velocity without acceleration?

If we assume that the stone rises and falls the same distance, the stone rises for 4 seconds. As the stone rises to its maximum height, its velocity decreases from 12 m/s to 0 m/s. Use the following equation to determine g. vf = vi – g * t

0 = 12 – g * 4, g = 3 m/s^2

Now we can use this value of g to determine the mass of the planet. Use the following equation.

Fg = G * M * m ÷ d^2

Fg is the weight of the stone

m * g = G * M * m ÷ d^2

g = G * M ÷ d^2

3 = G * M ÷ d^2

G = 6.67 * 10^-11

M = mass of planet

d = radius of the planet

3 = 6.67 * 10^-11 * M ÷ r^2

M = 3 * r^2 ÷ 6.67 * 10^-11

Use the following equation to determine the radius.

C = 2 * π * r

2 * 10^5 = 2 * π * r

r = 1 * 10^5/π ≈ 3.183098862 * 10^4 m

M = 3 * (1 * 10^5/π)^2 ÷ 6.67 * 10^-11 = 4.557174677 * 10^19 kg

Check:

3 = 6.67 * 10^-11 * 4.557174677 * 10^19 ÷ (1 * 10^5/π)^2

3 = 3.000001378

This proves that mass is correct.

Additional Details

Also the full question is this-

Your starship lands on a mysterious planet. You decide to throw a 2.50kg stone upward at 12m/s, which takes 8 seconds to land again. The circumference of the equator is 2.00E5. Find the mass of the planet and the time it would take to orbit the planet at 30,000 km above the surface.

Let’s determine the radius of the orbit. Radius = radius of planet + altitude

Let’s convert the altitude to meters by multiplying by 1000. A = 3 * 10^7 m

Radius = 1 * 10^5/π + 3 * 10^7 ≈ 3.003183099 * 10^7

The time to orbit is equal to the length of the obit divided by the velocity of the object. The length of the obit is equal to the circumference of the orbit.

Time = 2 * π * r ÷ v

Since the orbit is circular, the gravitational force is equal to the centripetal force

m * v^2 = G * M * m ÷ d^2

v^2 = G * M ÷ d^2

v = √(G * M ÷ d^2)

d = radius of the orbit

v = √(G * M ÷ r^2)

Time = 2 * π * r ÷ √(G * M ÷ r^2)

T^2 = 4 * π^2 * r^2 ÷ (G * M ÷ r^2)

T^2 = 4 * π^2 * r^3 ÷ (G * M)

r = 1 * 10^5/π + 3 * 10^7 ≈ 3.003183099 * 10^7

4 * π^2 * r^3 = 4 * π^2 * (3.003183099 * 10^7)^3 = 1.069313797 * 10^24

(G * M) = 6.67 * 10^-11 * 3 * (1 * 10^5/π)^2 ÷ 6.67 * 10^-11

(G * M) = 3 * (1 * 10^5/π)^2 = 3.039635509 * 10^9

T^2 = 1.069313797 * 10^24 ÷ 3.039635509 * 10^9

T = √(1.069313797 * 10^24 ÷ 3.039635509 * 10^9)

T = 1.875606938 * 10^7 seconds

Check:

T^2 = 4 * π^2 * r^3 ÷ (G * M)

(1.875606938 * 10^7)^2 = 4 * π^2 * (3.003183099 * 10^7)^3 ÷ 3 * (1 * 10^5/π)^2

3.517901386 * 10^14 = 1.069313797 * 10^24 ÷ 3.039635509 * 10^9

3.517901386 * 10^14 = 3.157901385 * 10^14

Since these answers are almost the same, I believe the period is correct.