A student initially at rest on a frictionless frozen pond throws a 1 kg hammer in one direction. After the throw the hammer moves off in one direction while the student moves off in the other direction. Which of the following correctly describes the above situation?
The answer is that the hammer will have the greater kinetic energy. Can anyone explain why?
They will both have the same linear momentum magnitude but in opposite directions, since momentum is conserved
mhammer Vhammer = mstudent Vstudent
so
Vh = (ms/mh)Vs
KEhammer = (1/2)mh [(ms Vs)/mh]^2
= (1/2)( ms^2/mh)Vs^2
but
Ke student = (1/2) ms Vs^2
well ms^2/mh >> ms
so the hammer has the big Ke
Certainly! According to Newton's third law of motion, for every action, there is an equal and opposite reaction. In this situation, when the student throws the hammer, they exert a force on the hammer in one direction. As a result, the hammer exerts an equal and opposite force on the student in the opposite direction.
Since the hammer is more massive than the student (1 kg vs the student's mass), the force exerted on the student will cause a smaller acceleration compared to the hammer. This is described by Newton's second law, which states that force is equal to mass multiplied by acceleration (F = m * a).
As a result, while both the hammer and the student experience the same magnitude of force (because of Newton's third law), the hammer will have a greater acceleration.
Now, the kinetic energy of an object is given by the equation KE = 0.5 * m * v^2, where m is the mass of the object and v is its velocity. Since the hammer has a greater acceleration, it will reach a higher velocity compared to the student.
Since the mass of the hammer and the student are the same (1 kg), and assuming they both started from rest, the hammer will have a greater kinetic energy.
Certainly! To understand why the hammer will have greater kinetic energy, let's analyze the situation using the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces are acting on it. In this case, since there is no friction or external force on the student and the hammer, the initial momentum of the system (before the throw) will be equal to the final momentum (after the throw).
Initially, both the student and the hammer are at rest, so their initial momentum is zero. After the throw, the hammer moves off in one direction, gaining momentum, while the student moves off in the opposite direction, also gaining momentum. According to the conservation of momentum, the magnitudes of these two momenta should be equal, but their directions are opposite.
Since momentum is defined as mass times velocity (p = mv), the product of mass and velocity for the hammer and the student will be equal in magnitude but opposite in direction. As momentum is a vector quantity, it takes into account both the magnitude and direction of the object's motion.
Now, let's consider kinetic energy. Kinetic energy is given by the equation KE = (1/2)mv^2, where m is the mass and v is the velocity of the object. Note that kinetic energy depends on both the mass and the square of velocity.
Since both the hammer and the student have the same mass (1 kg), the one with greater kinetic energy must have a higher velocity. Given that the hammer moves off in one direction while the student moves in the opposite direction, the velocity of the hammer must be greater.
Therefore, the hammer will have a greater kinetic energy because it has a higher velocity.