The Little Prince is a fictional character who lives on a very small planet as shown in the figure below. Suppose that the planet has a mass of 1.95 1013 kg and a radius of 1.06 103 m.
(a) How long would it take for an object to fall from rest a vertical distance of 1.16 m?
(b) Suppose the Little Prince throws a ball vertically upward, giving it an initial velocity of 1.00 m/s. What would be the maximum height reached by the ball? (HINT: Don't assume g to be constant.)
Incorrect: Your answer is incorrect.
1.16 m is tiny compared to the radius of 1.06*10^3 meters so I will assume g is constant for part (a)
m g = G m M/r^2
g = 6.67*10^-11 (1.95*10^13)/(1.06^2*10^6)
= 11.6*10^-4 m/s^2
d = (1/2)gt^2 =1.16
for part (b) the kinetic energy + potential energy at the bottom = potential energy at the top.
(1/2)m v^2 - G mM/r= -GmM/(r+h)
1/2 = GM [1/r - 1/(r+h)]
.5 = (6.67*10^-11)(1.95*10^13)[1/r - 1/(r+h)]
.0384*10^-2 = [1/r - 1/(r+h)]
.0384*10^-2 = 1/1.06*10^3 -1/(1.06*10^3+h)
-.0384*10^-2 + .000943 = 1/(1060+h)
.000559 = 1/(1060+h)
1060+h = 1789
h = 729 meters (huge due to tiny g)
To calculate the time taken for an object to fall from rest a vertical distance of 1.16 m on the planet, we can use the equation for the time of free fall:
t = √(2h/g)
Where:
t = time taken (unknown)
h = vertical distance (1.16 m)
g = acceleration due to gravity on the planet
To find the value of g, we can use the formula:
g = GM/r^2
Where:
G = universal gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = mass of the planet (1.95 x 10^13 kg)
r = radius of the planet (1.06 x 10^3 m)
Let's calculate it step by step:
(a) Time taken for an object to fall from rest a vertical distance of 1.16 m:
Step 1: Calculate the gravity on the planet:
g = (6.67430 x 10^-11) x (1.95 x 10^13) / (1.06 x 10^3)^2
Step 2: Substitute the values into the time of free fall equation:
t = √(2 x 1.16 / g)
Step 3: Calculate the time taken:
t = √(2 x 1.16 / g)
Now, to calculate the maximum height reached by a ball thrown vertically upward by the Little Prince with an initial velocity of 1.00 m/s, we need to find the maximum height (H).
To do that, we can use the equation:
H = h + (v^2 / (2g))
Where:
H = maximum height (unknown)
h = initial height (0 m)
v = initial velocity (1.00 m/s)
g = acceleration due to gravity on the planet
Let's calculate it step by step:
(b) Maximum height reached by the ball:
Step 1: Calculate the gravity on the planet:
g = (6.67430 x 10^-11) x (1.95 x 10^13) / (1.06 x 10^3)^2
Step 2: Substitute the values into the maximum height equation:
H = 0 + (1.00^2 / (2 x g))
Step 3: Calculate the maximum height:
H = (1.00^2 / (2 x g))