magine a landing craft approaching the surface of Callisto, one of Jupiter's moons. If the engine provides an upward force (thrust) of 3270 N, the craft descends at constant speed; if the engine provides only 2190 N, the craft accelerates downward at 40 m/s2. What is the weight of the landing craft in the vicinity of Callisto's surface? What is the mass of the craft? What is the magnitude of the free-fall acceleration near the surface of Callisto?

I will be happy to critique your thinking on this. Look at the other two problems for closely related examples.

Wow, landing on Jupiter's moon Callisto? That's out of this world! Let's crunch some numbers, shall we?

First, when the engine provides an upward thrust of 3270 N, the craft descends at constant speed. This means the weight of the landing craft is exactly balanced by the upward force from the engine. So, the weight of the craft is 3270 N.

Now, what about when the engine provides only 2190 N? In this case, the craft accelerates downward at 40 m/s^2. To calculate the net force acting on the craft, we need to subtract the upward force from the engine (2190 N) from the weight of the craft. So, the net force is the weight (3270 N) minus the upward force (2190 N), which gives us 1080 N.

We know that force equals mass times acceleration (F = ma). Plugging in the values we have, 1080 N = mass x 40 m/s^2. To find the mass of the craft, we can rearrange the equation to solve for mass: mass = 1080 N / 40 m/s^2. That gives us a mass of 27 kg.

Finally, the magnitude of the free-fall acceleration near the surface of Callisto is approximately 40 m/s^2, given the craft accelerated downward at that rate.

I hope my calculations made you smile!

Step 1: Calculate the weight of the landing craft in the vicinity of Callisto's surface.

The weight of an object can be calculated using the formula:

Weight = Mass x Acceleration due to gravity

Since the craft is in the vicinity of Callisto's surface, we need to determine the acceleration due to gravity on Callisto.

Step 2: Calculate the mass of the landing craft.
To find the mass, we need to rearrange the equation from Step 1:

Mass = Weight / Acceleration due to gravity

Step 3: Find the magnitude of the free-fall acceleration near the surface of Callisto.
The magnitude of the free-fall acceleration near the surface of Callisto would be equal to the acceleration due to gravity.

Let's proceed with the calculations:

Step 1: Calculate the weight of the landing craft.
Given:
Thrust (upward force) at constant speed = 3270 N
Thrust (upward force) with downward acceleration = 2190 N
Acceleration of descent = 40 m/s^2

Using the first condition (descending at constant speed):
Weight = Thrust = 3270 N

Using the second condition (acceleration downward at 40 m/s^2):
Weight - Thrust = mass x acceleration
Weight - 2190 N = mass x (-40 m/s^2)

Step 2: Calculate the mass of the landing craft.
Weight - 2190 N = mass x (-40 m/s^2)
Weight = mass x (-40 m/s^2) + 2190 N

Since we already calculated the weight in the first condition as 3270 N, we can now substitute this value into the equation:
3270 N = mass x (-40 m/s^2) + 2190 N

Simplifying the equation:
3270 N - 2190 N = mass x (-40 m/s^2)
1080 N = -40 m/s^2 x mass

Dividing both sides by -40 m/s^2:
mass = 1080 N / (-40 m/s^2)

Step 3: Find the magnitude of the free-fall acceleration near the surface of Callisto.
The magnitude of the free-fall acceleration near the surface of Callisto is the same as the acceleration due to gravity. Unfortunately, the given information does not explicitly state the acceleration due to gravity on Callisto.

Therefore, in order to fully answer this question, we require additional information about the acceleration due to gravity on Callisto.

To find the weight of the landing craft in the vicinity of Callisto's surface, we can use the equation:

Weight = Mass * Acceleration due to gravity

From the information given, we can deduce that when the engine provides an upward force (thrust) of 3270 N, the craft descends at a constant speed. This means that the net force acting on the craft is equal to zero.

The net force acting on an object is given by the equation:

Net Force = Thrust - Weight

Since the craft descends at a constant speed, the net force is zero. Therefore:

Thrust - Weight = 0

Rearranging the equation, we have:

Weight = Thrust

Therefore, the weight of the landing craft in the vicinity of Callisto's surface is 3270 N.

To find the mass of the craft, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

Net Force = Mass * Acceleration

From the information given, when the engine provides only 2190 N of thrust, the craft accelerates downward at 40 m/s^2. So, we can write:

Net Force = 2190 N
Acceleration = 40 m/s^2

Plugging these values into the equation, we have:

2190 N = Mass * 40 m/s^2

Simplifying the equation, we find:

Mass = 2190 N / 40 m/s^2

Mass = 54.75 kg

Therefore, the mass of the landing craft is approximately 54.75 kg.

To find the magnitude of the free-fall acceleration near the surface of Callisto, we can use Newton's law of gravitation:

F = G * (m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the objects (in this case, the craft and Callisto), and r is the distance between the centers of the objects.

In the vicinity of Callisto's surface, the craft and Callisto are very close, so we can approximate the distance between their centers as the radius of Callisto.

The weight of the craft, which we found earlier, is equal to the gravitational force acting on it. Therefore:

Weight = F

Weight = G * (m1 * m2) / r^2

Substituting the weight of the craft, the mass of the craft, and the radius of Callisto, we have:

3270 N = G * (54.75 kg * m2) / (radius of Callisto)^2

Since we're solving for the free-fall acceleration near the surface of Callisto, we can rearrange the equation:

Acceleration due to gravity = G * (m2 / (radius of Callisto)^2)

Now, we need to look up the value of the gravitational constant (G) and the radius of Callisto to calculate the acceleration due to gravity accurately. The value of G is approximately 6.67430 x 10^-11 m^3/kg/s^2, and the radius of Callisto is about 2,410,000 meters.

Plugging these values into the equation, we find:

Acceleration due to gravity = (6.67430 x 10^-11 m^3/kg/s^2) * (54.75 kg) / (2,410,000 meters)^2

Calculating this expression, we get:

Acceleration due to gravity ≈ 1.2369 m/s^2

Therefore, the magnitude of the free-fall acceleration near the surface of Callisto is approximately 1.2369 m/s^2.