Question

1. is it possible for the sum of kinetic energy and potential energy of an object to change without work having been done on the object? explain.

a. no, because the work - energy theorem states that work done on an object is equal to the change in kinetic energy, and change in ke requires a change in velocity. it is assumed that mass is constant.
b. no, because the work - energy theorem states that work done on an object is equal to the sum of kinetic energy, and the change in ke requires a change in displacement. it is assumed that mass is constant.
c. yes, because the work - energy theorem states that work done on an object is equal to the change in kinetic energy, and change in ke requires a change in velocity. it is assumed that mass is constant.
d. yes, because the work - energy theorem states that work done on an object is equal to the sum of kinetic energy, and the change in ke requires a change in displacement. it is assumed that mass is constant.

1. define work for one - dimensional motion.

a. work is defined as the ratio of the force over the distance.
b. work is defined as the sum of the force and the distance.
c. work is defined as the square of the force over the

Explanation:

Question 1

The work - energy theorem states that the work done on an object ($W$) is equal to the change in its kinetic energy ($\Delta KE$), i.e., $W=\Delta KE$. Kinetic energy is given by $KE = \frac{1}{2}mv^{2}$, so a change in kinetic energy requires a change in velocity (since mass $m$ is assumed constant). For the sum of kinetic and potential energy (mechanical energy) to change, either KE or PE (or both) must change. But from the work - energy theorem, a change in KE (and thus a change in the sum of KE and PE, if PE also changes or not) requires work to be done. So the answer should be option a as it correctly states the work - energy theorem and the relationship between work, KE, and velocity.

The correct definition of work for one - dimensional motion is $W = Fd\cos\theta$, where $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors. In one - dimensional motion (when the force and displacement are in the same or opposite direction, $\cos\theta=\pm1$), work is the product of force and displacement. None of the given options (a: ratio of force over distance, b: sum of force and distance, c: square of force over distance) is correct. But if we assume there is a typo and maybe the intended correct option was not provided, or if we consider the options given, all are incorrect. However, if we have to choose from the given wrong options, there is a mistake. But following the problem's context, if we assume that maybe the options were miswritten and the correct concept is that work is force times displacement, but among the given options, none is correct. But since the problem asks to choose from a, b, c, and all are wrong, but perhaps there was a mistake in the problem. If we strictly go by the given options, there is no correct answer, but if we assume a misprint and the intended definition was force times displacement (which is not among the options), but as per the given options, all are incorrect.

Answer:

a. No, because the work - energy theorem states that work done on an object is equal to the change in kinetic energy, and change in KE requires a change in velocity. It is assumed that mass is constant.

Question 2