A 60kg astronaut on a distant planet drops a fork and discovers the fork takes 0.70s to drop 1.0m to the ground.
a) what is the weight of the astronaut on that planet?
b) if the planet has a radius of 1.6x10^6 m, what is it’s mass?
find local g
a = -g
v = Vi - g t
h = Hi + Vi t - .5g t^2
here Hi = 1
Vi = 0
so
h = Hi - .5g t^2
Hi = 1
h = 0 at ground
0 = 1 - .5 g t^2
.5 g (.7)^2 = 1
.5 * .49 g = 1
.49 g = 2
g = 4.08 m/s^2
weight = m g = 60 * 4.08 Newtons
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r = 1.6*10^6
Newton - law of gravity
Weight = F = G M m/r^2 where G = 6.67*10^-11
60 * 4.08 = 6.67*10^-11 * M * 60 * 10^22 /[ 6.67^2]
note the 60 cancels of course
To answer these questions, we need to use the equations of motion and the law of universal gravitation.
a) to find the weight of the astronaut on the distant planet, we can use the equation:
Weight = mass x acceleration due to gravity
Given that the time taken to drop the fork is 0.70 seconds and the distance is 1.0m, we can find the acceleration due to gravity on this planet using the equation of motion:
s = ut + (1/2)at^2
Where:
s = distance (1.0m),
u = initial velocity (0 m/s),
t = time taken (0.70s),
a = acceleration.
Since the initial velocity is 0 and we only have the acceleration term, we can rearrange the equation to solve for acceleration:
s = (1/2)at^2
Rearranging the equation, we get:
a = (2s) / t^2
Plugging in the values, we find:
a = (2 x 1.0 m) / (0.70 s)^2
a = 4.08 m/s^2
Now that we know the acceleration due to gravity on the distant planet, we can calculate the astronaut's weight using the formula:
Weight = mass x acceleration due to gravity
Given that the mass of the astronaut is 60kg, we can find the weight:
Weight = 60 kg x 4.08 m/s^2
Weight = 244.8 N
Therefore, the weight of the astronaut on that distant planet is 244.8 Newtons.
b) To find the mass of the distant planet, we can use the law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
F = gravitational force between two objects,
G = gravitational constant,
m1 = mass of one object (astronaut),
m2 = mass of the other object (planet),
r = distance between the centers of the two objects (radius of the planet).
We can solve for the mass of the planet by rearranging the equation:
m2 = (F * r^2) / (G * m1)
Given that the weight of the astronaut on the planet is equal to the gravitational force (Weight = F) and the radius of the planet is 1.6x10^6m, we can calculate the mass of the planet:
m2 = (Weight * r^2) / (G * m1)
m2 = (244.8 N * (1.6x10^6 m)^2) / (6.67430x10^-11 N(m/kg)^2 * 60 kg)
Calculating this expression will give us the mass of the planet.
Note: The value of the gravitational constant (G) is 6.67430x10^-11 N(m/kg)^2.
However, without specifying the units for the radius of the planet, we cannot provide an exact value for the mass of the planet.