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).
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.
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