Conundrums on energy and graphs

[aspodkfpo]

Homework Statement

Multiple questions that I have regarding concepts.

Relevant Equations

KE=1/2mv^2, W=fdcos(theta), W=fs

1. I wanted to clarify if Kinetic energy is always positive. Since KE = 1/2 mv^2, and m and the square of v is positive. I assume as such.

2. Given that I have a scenario where an object which was traveling at a positive velocity in a certain direction (we take this direction as positive), reverses the direction and travels at a negative velocity similar in magnitude. The KE final and initial would be equal due to KE always being positive. Now, the change in KE would be 0, since both values are equal positive values. However, that is not the case as W = fdcos(theta). And clearly, there must be a force enacted to change the velocity of the object. Assume there is a distance present over which the force acts. Now, the work should be negative as cos(theta) is negative.

Given this, I don't understand what I am supposed to do when writing KE or W. If KE is negative relative to something else, would I write KE = - 1/2 mv^2?

3. W=fdcos(theta) or W=Fs. In the cos theta version, are f and d magnitude/s, vector/s or scalar/s? In the w=fs version, I have the same question.

4. I read somewhere that scalars are simply values. In that case, are scalars magnitudes with signs, rather than just magnitudes? Or is a magnitude being an absolute value a misconception?

5. Say that I have a velocity time graph, and the velocity changes at a certain point. These points of change can not be differentiated can they? Would the end point's slope be undefined as well?

 

[PhanthomJay]

Yes kinetic energy is always positive and it is a scalar quantity. Work is also a scalar, and can be negative or positive. You can’t differentiate at a discontinuity in the curve, but in actuality, it is not really a discontinuity because you can’t have an instantaneous change in acceleration, so rather, that point is actually a short curve as the acceleration changes to its new value in a small finite period of time and distance.
Oh you are new here so welcome to PF!

 

[aspodkfpo]

Yes kinetic energy is always positive and it is a scalar quantity. Work is also a scalar, and can be negative or positive. You can’t differentiate at a discontinuity in the curve, but in actuality, it is not really a discontinuity because you can’t have an instantaneous change in acceleration, so rather, that point is actually a short curve as the acceleration changes to its new value in a small finite period of time and distance.
Oh you are new here so welcome to PF!

What about question 3 and question 4? I've figured out my problem in q2.

 

[PhanthomJay]

3. W=fdcos(theta) or W=Fs. In the cos theta version, are f and d magnitude/s, vector/s or scalar/s? In the w=fs version, I have the same question.

In the first, F and d are the magnitudes of the vectors F and d, respectively, and work is a scalar.in the second, it is W = F.s, the dot product of 2 vectors, which is fscostheta, the same result.

4. I read somewhere that scalars are simply values. In that case, are scalars magnitudes with signs, rather than just magnitudes? Or is a magnitude being an absolute value a misconception?

scalars do not have direction. They can be plus or minus, but the plus or minus signs do not indicate direction. Like temperature is a scalar and it can be above or below 0 degrees C. The confusion may be in the use of the word ‘magnitude’. The magnitude of a vector is it’s absolute value and is a positive number, but the magnitude of a scalar can be positive or negative. Kinetic energy is a scalar always positive because of the squared term, but potential energy is also a scalar which can be positive or negative.

 

[PeroK]

2. Given that I have a scenario where an object which was traveling at a positive velocity in a certain direction (we take this direction as positive), reverses the direction and travels at a negative velocity similar in magnitude. The KE final and initial would be equal due to KE always being positive. Now, the change in KE would be 0, since both values are equal positive values. However, that is not the case as W = fdcos(theta). And clearly, there must be a force enacted to change the velocity of the object. Assume there is a distance present over which the force acts. Now, the work should be negative as cos(theta) is negative.

You could think about an object moving in one direction, impacting a spring, compressing the spring until it stops, then being accelerated by the spring back in the opposite direction.

This system starts and ends with the same energy, all the KE of the moving object. The total work done by the spring on the object is zero. Can you see why?

 

[jbriggs444]

4. I read somewhere that scalars are simply values. In that case, are scalars magnitudes with signs, rather than just magnitudes? Or is a magnitude being an absolute value a misconception?

If you ever get to the point of studying linear algebra, the notions of vectors and of scalars are given a much more formal footing.

Quoting from the article linked above:

"A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars."

For now (until you get to vector spaces without norms or scalars that are not real numbers):

Scalars can be negative. Magnitudes are always positive (or zero).