While visiting Planet Physics, you toss a rock straight up at 13.6 m/s and catch it 3.50 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 340.0 minutes. What is the radius (in m) of the planet?
What is the mass (in kg) of the planet?
3.5/2 = 1.75 seconds to peak height
v = Vi - g t
0 = 13.6 - g (1.75) solve for local g on surface. At 2 r it is g/4
340 min = 340 * 60 seconds = T
m * centripetal acceleration = m (g/4) at 2 radii from center
m v^2/2r = = m g/4
v^2/r = g/2
but we know T
v T = 2 pi (2r) = 4 pi r
v = 4 pi r/T
v^2 = 16 pi^2 r^2/T^2
v^2/r = (16 pi^2/T^2) r
so in the end
r = (g/2)/(16 pi^2/T^2)
To find the radius of the planet, we can use the information given about the rock's motion.
The initial velocity of the rock when it was tossed straight up is 13.6 m/s. The total time it takes for the rock to go up and come back down is 3.50 s.
Since the rock goes up and then comes back down, the total time of flight is twice the time it took to reach the maximum height. Therefore, the time it took for the rock to reach maximum height is 3.50 s divided by 2, which is 1.75 s.
Using the kinematic equation for vertical motion:
v = u + gt
where:
v = final velocity (0 m/s when it reaches maximum height)
u = initial velocity (13.6 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
We can solve for the time it took to reach maximum height (t) using the equation:
0 = 13.6 - 9.8t
Rearranging the equation, we get:
9.8t = 13.6
Dividing both sides of the equation by 9.8, we find:
t = 13.6 / 9.8
t = 1.3878 s
So, the time it took for the rock to reach maximum height is approximately 1.3878 s.
Now, we can calculate the height reached by the rock using the equation:
s = ut + (1/2)gt^2
where:
s = height reached
u = initial velocity (13.6 m/s)
t = time to reach maximum height (1.3878 s)
g = acceleration due to gravity (-9.8 m/s^2)
Plugging in the values, we get:
s = 13.6 * 1.3878 + (1/2) * (-9.8) * (1.3878)^2
Simplifying this equation, we find:
s = 9.47 m
Since the height reached by the rock is equal to the radius of the planet, the radius of the planet is 9.47 m.
To find the mass of the planet, we need to use the information about the cruise ship's orbit.
The cruise ship orbits the planet at an altitude equal to the planet's radius every 340.0 minutes.
Using Kepler's Third Law of Planetary Motion, we can relate the orbital period (T) of the cruise ship and the radius (R) of the planet:
T^2 = k * R^3
where:
T = orbital period of the cruise ship in seconds (340.0 minutes = 340.0 * 60 seconds)
R = radius of the planet
k = constant (depends on the gravitational constant and the mass of the planet)
We need to rearrange the equation to solve for the mass of the planet (M):
M = (4π^2 * R^3) / (kT^2)
To find the value of k, we can use the known values of the gravitational constant (G) and the mass of the Sun (M_sun):
k = (G * M_sun) / (4π^2)
Plugging in the known values:
G = 6.67430 x 10^-11 m^3 kg^-1 s^-2
M_sun = 1.989 x 10^30 kg
Using this information, we can calculate k:
k = (6.67430 x 10^-11 * 1.989 x 10^30) / (4π^2)
k = 1.8548 x 10^-19
Now, we can calculate the mass of the planet:
M = (4π^2 * R^3) / (kT^2)
Plugging in the known values:
T = 340.0 * 60 s
R = 9.47 m
k = 1.8548 x 10^-19
M = (4π^2 * (9.47)^3) / (1.8548 x 10^-19 * (340.0 * 60)^2)
Calculating this expression:
M ≈ 2.27 x 10^24 kg
Therefore, the mass of the planet is approximately 2.27 x 10^24 kg.
To find the radius of the planet, we can start by using the information given about the rock's motion.
We know that the initial velocity (u) of the rock is 13.6 m/s and the time it takes to reach its highest point and fall back down (t) is 3.50 s. In this case, the vertical displacement (h) is zero because the rock starts and ends at the same height.
To calculate the time it takes for the rock to reach its highest point, we can use the equations of motion. The equation for calculating displacement (s) is:
s = ut + 1/2 * a * t^2
Since h = 0 (no vertical displacement), the equation becomes:
0 = u * t + 1/2 * (-g) * t^2
Simplifying this equation, we get:
13.6 * 3.50 - 1/2 * 9.8 * (3.50^2) = 0
Solving this equation will give us the value of g, the acceleration due to gravity on the planet. Once we know the value of g, we can use it to find the radius of the planet.
Now, to find the mass of the planet, we will use the information given about the cruise ship's orbit.
We know that the cruise ship orbits at an altitude equal to the planet's radius every 340.0 minutes. This gives us the orbital period (T) of the cruise ship.
Using Kepler's third law of planetary motion, we can relate the period of orbit (T) to the radius (r) and the mass (M) of the planet as follows:
T^2 = (4π^2 * r^3) / (G * M)
In this equation, G is the gravitational constant.
By solving this equation for the mass (M) of the planet, we can find the mass in kilograms.
In summary, to find the radius of the planet, we need to solve the equation involving the rock's motion, and to find the mass of the planet, we need to solve the equation involving the cruise ship's orbit.