A scene depicting the physics concept explained in a question related to the kinetic energy of a projectile escaping a planet's gravitational field. Display a visual of a planet with radius R having a floating rocket about to escape its gravitational field, with the elements of mass m, kinetic energy K, and gravitational acceleration g clearly defined. Also, depict the second scenario, a system of particles with total external forces zero, encompassing different particles showcasing the concepts of kinetic energy, potential energy, mechanical energy, and linear momentum, to illustrate the point.

Please help...

Let g be the acceleration due to gravity at the surface of a planet of radius R. Which of the following is a dimensionally correct formula for the minimum kinetic energy K that a projectile of mass m must have at the planet's surface if the projectile is to escape from the planet's gravitational field?
A) K=sqt.(gR)
B) K=mgR
C) K=mg/R
D) K=m*sqt.(g/R)
E K=gR

(I thought the answer should be K=mgR/2, but the key is B)... )

The sum of all the extenral forces on a system of particles is zero. Which of the following must be true of the system?
A)the total mechanical energy is constant
B)total potential energy is constant
C)total kinetic energy is constant
D)total linear momentum is constant
E) it is in static equilibrium

I am so confused of these kind of question. How can I do these kind of problem? And when I see a problem, how can I figure out which method to use, momentum conservation or energy conservation, is easier? Is any characteristics?

Thanks!

A) The force acting on mass m at distance r is
F = -mg (R^2/r^2)
The potential energy is minuus the r integral of that, which is PE = mg R^2/r
Starting at distance r=R, the kinetic energy neecded to have a zero velocity at r=infinity is
PE @ (r=R) = mgR

B) All statements are true. With no net force, the total kinetic energy and the momentum cannot change. Total mechanical energy (PE + KE) is conserved, always (since there is no friction). If total E and KE are constant, so is PE.

I don't quite understand the first one. How did you come up with the first equation?
What I did was using mg=mv^2/R, then K=(mv^2)/2

And the answer key said the second one is D) correct...

You then found te kinetic energy for an object in orbit. The energy needed to escape is twice this value. But that factor 2 is irrelevant in this problem because you only need to select the formula which is dimensionally correct (and you can thus ignore dimensionless factors of 2).

In fact, the method you used (to write down an irrelevant equation which nevertheless has the right dimensions) is in many cases faster than the traditional method of substituting the dimensions in terms of meters, seconds kilograms in the formulae to select the one which has the dimensions of energy (kg m^2/s^2).

You still need to show that the other formulae are not dimensionally correct, you do that by dividing the other formula by the one you know is correct and show that the result is not dimensionless.

In the second problem drwls, looked only at the center of mass energies. The total kinetic energy is not constant because there can be internal foces between the paticles which can cause the particles to accelerate relative to each other. The total potential energy will then also change.

The total mechanical energy does not need to be conserved, you can e.g. think of inelastic collisions between the particles.

To have static equilibrium the force on each particle separately must be zero, not just the total force. In case of continuous objects you may be familiar with the condition that you need a zero torque in addition to zero total force...

I do not agree with the some of statements of Count Iblis regarding my answer. My comments referred to the total potential and kinetic energy of the SYSTEM of particles. I believe that was the intent of your question. I neglected the possibility of heat transfer, which could add to total system mechanical energy.

Regarding the first question, only (B) has the dimensions of energy,
kg* m/s^2*m = Joules,
so none of the other answers could be right. That could be determined without actually solving the problem. (B) is also numerically the correct answer, and I tried to explain why.

Well, it seems like there was a bit of confusion with the first question. But hey, don't worry, it happens to the best of us. As for figuring out which method to use, it really depends on the problem. Sometimes momentum conservation is easier, sometimes energy conservation is easier, and sometimes both can be used together. There's no one-size-fits-all answer.

The key is to carefully read the problem and analyze what information is given and what is being asked. Look for clues in the problem statement, such as references to forces or velocities, as they can suggest which method to use. And if in doubt, try both approaches and see which one leads to the correct answer.

But remember, even if you get a bit confused along the way, just keep a positive attitude and keep on clowning around. After all, laughter is the best way to solve any problem!

To understand the first equation, let's break it down step-by-step:

1. First, recall that the force acting on an object in the gravitational field near the planet's surface is given by Newton's law of universal gravitation: F = mg, where m is the mass of the object and g is the acceleration due to gravity.
2. However, g can vary with distance from the center of the planet. Near the planet's surface, g can be approximately considered constant.
3. Now, let's consider a small mass element of mass dm at a distance r from the center of the planet. The force acting on this mass element is given by F = -G (Mm/r^2), where G is the gravitational constant and M is the mass of the planet.
4. Now, the mass of the small element dm can be written as dm = (4πr^2ρ)dr, where ρ is the density of the planet.
5. Substituting the value of dm in the force equation, we get F = -(4πGρM / r) r^2 dr = -g(r) dm.
6. Comparing this equation with F = mg, we find that g(r) = 4πGρ r / 3M.
7. Now, let's consider the force acting on a mass m at the surface of the planet, which has a radius R. The force acting on this mass is given by F = -mg(R) = -4πGρ R^2 m / 3M.
8. The negative sign indicates that the force is directed towards the center of the planet.
9. To calculate the potential energy, we integrate the force over the distance r from the surface to infinity: PE = ∫ -4πGρ R^2 m / 3M dr = -4πGρ R^2 m / 3M ∫ dr = -4πGρ R^2 m / 3M (r + C), where C is the constant of integration.
10. Evaluating the integral from r = R to r = ∞, we find PE = -4πGρ R^2 m / 3M (∞ - R) = -4πGρ R^2 m / 3M * (-R) = 4πGρ R^3 m / 3M.
11. The minimum kinetic energy (K) required for the projectile to escape from the gravitational field is equal to the magnitude of the potential energy: K = -PE = -4πGρ R^3 m / 3M.
12. Taking the magnitude, we find K = 4πGρ R^3 m / 3M.

Therefore, the correct answer is B) K = mgR, which matches the derived equation 4πGρ R^3 m / 3M when we assume a constant density for the planet.

Regarding your second question about selecting the right method (momentum conservation or energy conservation), it depends on the specific problem and the information given. Here are some general guidelines:

1. Momentum conservation is useful when dealing with collisions or interactions between objects. It focuses on the total momentum before and after the interaction.
2. Energy conservation is useful when dealing with potential energy changes (such as gravitational potential energy), kinetic energy changes, or work done by non-conservative forces. It focuses on the total energy before and after the process.

To determine which method to use, consider the following:
- If the problem involves collisions or interactions between objects, start with momentum conservation.
- If the problem involves potential energy changes, kinetic energy changes, or work done by non-conservative forces, start with energy conservation.

However, it's important to note that in some cases, both methods may need to be used together to fully analyze a problem. It's always helpful to analyze the given information and try to understand the underlying physics principles involved.

I hope this helps! Let me know if you have any further questions.

To answer the first question, we can analyze the dimensions of the given formulas and determine which one has the correct dimensions of energy (kg m^2/s^2).

Option A) K = sqrt(gR)
Breaking it down:
sqrt(gR) = sqrt(m/s^2 * m) = sqrt((kg m/s^2) * m) = sqrt(kg m^2/s^2) = sqrt(Joules) = not the correct dimension of energy.

Option C) K = mg/R
Breaking it down:
mg/R = (kg * m/s^2) / m = kg/s^2 = not the correct dimension of energy.

Option D) K = m*sqrt(g/R)
Breaking it down:
m*sqrt(g/R) = (kg)*(sqrt(m/s^2)/m) = sqrt(kg m^2/s^2) = sqrt(Joules) = not the correct dimension of energy.

Option E) K = gR
Breaking it down:
gR = (m/s^2)*(m) = kg m^2/s^2 = the correct dimension of energy.

Therefore, the dimensionally correct formula for the minimum kinetic energy K is option E) K = gR.

As for knowing which method to use in these types of problems (momentum conservation or energy conservation), it depends on the context and what quantities are given or asked for in the problem.

If you are given information about forces, velocities, or masses of objects, momentum conservation might be a good approach to analyze the system. Momentum is conserved if the net external force acting on the system is zero.

If you are given information about potential energy, forces due to gravity, or heights of objects, energy conservation might be a good approach to analyze the system. Energy is conserved if there is no non-conservative forces (like friction), and you can track the changes in potential energy and kinetic energy of the system.

It's also worth noting that in some cases, both momentum conservation and energy conservation can be applied together to analyze a system.

In general, it's a good idea to carefully read the problem, identify the given information and what you are asked to solve for, and then choose the appropriate method (momentum conservation or energy conservation) based on what will help you solve the problem most effectively.

The minimum kinetic energy that a projectile must have at the surface of a planet in order to escape from the planet's gravitational field is given by the formula:

K = (1/2) * m * v^2

where v is the escape velocity of the projectile. The escape velocity is the minimum velocity that a projectile must have at the surface of a planet in order to escape from the planet's gravitational field.

The escape velocity of a projectile on a planet of radius R and mass M is given by the formula:

v = sqrt(2GM/R)

where G is the gravitational constant and M is the mass of the planet.

Therefore, the correct answer is (D) K=m*sqt.(g/R).