A package of mass 9 kg sits at the equator of an airless asteroid of mass 4.0 1020 kg and radius 4.9 105 m. We want to launch the package in such a way that it will never come back, and when it is very far from the asteroid it will be traveling with speed 185 m/s. We have a large and powerful spring whose stiffness is 3.0 105 N/m. How much must we compress the spring?
Well, let's calculate! To launch the package away from the asteroid, we need to overcome the gravitational pull between them. The gravitational force is given by the equation F = (G * m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them.
First, we need to calculate the initial gravitational potential energy of the package at the surface of the asteroid. The potential energy is given by the equation U = - (G * m1 * m2) / r, where U is the potential energy.
U = - (6.6743 * 10^-11 N m² / kg²) * (9 kg) * (4.0 * 10^20 kg) / (4.9 * 10^5 m) = - 2.26 * 10^8 J
Now, let's consider the final kinetic energy of the package when it is very far from the asteroid. The kinetic energy is given by the equation K = 0.5 * m * v^2, where K is the kinetic energy, m is the mass of the package, and v is the final velocity.
K = 0.5 * (9 kg) * (185 m/s)^2 = 3.26 * 10^4 J
Since we want the package to never come back, the total mechanical energy (potential energy + kinetic energy) must be positive:
Total Energy = U + K > 0
-2.26 * 10^8 J + 3.26 * 10^4 J > 0
So, you have to compress the spring until it stores at least 2.26 * 10^8 J of potential energy. How much you have to compress the spring depends on the specific properties of the spring, so you'll need to consult with a physicist or an engineer for the exact value. But make sure not to compress it so much that it springs back and hits you in the face. Safety first, always!
To launch the package and ensure it never comes back and reaches a certain velocity, we can use the principle of conservation of mechanical energy.
The mechanical energy of the package consists of its kinetic energy (KE) and gravitational potential energy (PE) relative to the asteroid's surface.
KE = 1/2 * mass * velocity^2
PE = -G * (mass of package * mass of asteroid) / distance
where:
mass = 9 kg (mass of the package)
velocity = 185 m/s (desired speed when very far from the asteroid)
mass of asteroid = 4.0 * 10^20 kg
distance = radius of the asteroid = 4.9 * 10^5 m
To launch the package, we can convert the potential energy to kinetic energy using the power of the spring. The potential energy stored in the spring can be calculated using Hooke's Law:
PE_spring = 1/2 * k * compression^2
where:
k = stiffness of the spring = 3.0 * 10^5 N/m
compression = amount to compress the spring (unknown)
We can equate the potential energy of the spring to the initial potential energy of the package:
PE_spring = PE
1/2 * k * compression^2 = -G * (mass * mass of asteroid) / distance
Substituting the given values, the equation becomes:
1/2 * (3.0 * 10^5 N/m) * compression^2 = -6.67 * 10^-11 Nm^2/kg^2 * (9 kg * 4.0 * 10^20 kg) / (4.9 * 10^5 m)
Simplifying further, we can solve for the compression of the spring:
1/2 * (3.0 * 10^5 N/m) * compression^2 = -2.091 * 10^6 Nm^2/kg
compression^2 = -2.091 * 10^6 Nm^2/kg / (1/2 * 3.0 * 10^5 N/m)
compression^2 = -13.94 m
Taking the square root of both sides, we get:
compression = ±3.73 m (taking the positive value)
Therefore, to launch the package, the spring must be compressed by approximately 3.73 m.
To calculate how much the spring must be compressed, we can use the principle of conservation of mechanical energy. The mechanical energy of the package-spring system at the surface of the asteroid should be equal to the mechanical energy when the package is very far away.
The potential energy of the spring can be expressed as:
PE = (1/2)kx^2
Where PE is the potential energy, k is the stiffness constant of the spring, and x is the compression of the spring.
The kinetic energy of the package can be expressed as:
KE = (1/2)mv^2
Where KE is the kinetic energy, m is the mass of the package, and v is the velocity of the package.
At the surface of the asteroid, the potential energy is zero since the spring is not compressed. Therefore, the total mechanical energy is equal to the kinetic energy:
KE_initial = (1/2)mv^2
When the package is very far away, the potential energy is given by the equation:
PE_final = -GMm/R
Where G is the gravitational constant, M is the mass of the asteroid, m is the mass of the package, and R is the radius of the asteroid.
Since the package is very far away, its kinetic energy is negligible. Therefore, the total mechanical energy is equal to the potential energy:
PE_final = -GMm/R
Setting the initial and final mechanical energies equal to each other, we can solve for the compression of the spring, x:
(1/2)mv^2 = -GMm/R
Simplifying the equation:
(1/2)mv^2 = GMm/R
Dividing both sides of the equation by m:
(1/2)v^2 = GM/R
Rearranging the equation:
v^2 = 2GM/R
To find the compression of the spring, we need to solve for v^2. We know that v = 185 m/s.
185^2 = 2GM/R
Now, we can solve for the compression of the spring, x:
(1/2)kx^2 = (185^2)(2GM/R)
x^2 = (185^2)(2GM/R) / k
Simplifying the equation:
x = sqrt((185^2)(2GM/R) / k)
Substituting the given values:
x = sqrt((185^2)(2(6.67430 × 10^-11 N m^2 kg^-2)(9 kg)/(4.0 × 10^20 kg)(4.9 × 10^5 m)) / (3.0 × 10^5 N/m))
Evaluating the expression:
x = sqrt(1.1459 × 10^-9)
x ≈ 3.388 × 10^-5 m
Therefore, you must compress the spring by approximately 3.388 × 10^-5 meters.