Professor added a twist to the conservation of energy problem.

[Randall]

Homework Statement

The force on a particle, acting along the x axis, varies as shown in the figure below. (a) Determine the work done by this force to move along the x-axis from x=0.0 to x = 10.0m and (b) from x=0.0 to x=15.0m and (c) what is the speed of the particle at each slope change and x-intercept.

(graph of force applied in the x-direction vs distance attached - please ignore all pencil markings, those are a part of a different discussion).

Homework Equations

W=F x d (work = force x distance)
W=PE + KE (work = potential energy + kinetic energy)
PE = mgh (potential energy = mass times gravity times height)
KE = (1/2)mv^2 (kinetic energy = 1/2 times mass times velocity squared
PE initial + KE initial = PE final + KE final (conservation of energy)

The Attempt at a Solution

For part (a) and (b), I found the areas under or above the red line - that was fairly straightforward, BUT the professor added in his own part (c), that being to find the speed of the particle at each slope change and x-intercept. I am certain the solution to part (c) has something to do with the conservation of energy. I thought maybe the total work done on and by the particle would cancel each other out, and therefore I can set the conservation of energy equation to zero and solve for particle speed, and the mass (which is not given) would simple cancel out, but since the amount of work done on and by the particle are different areas on the graph, they don't cancel each other out. Therefore, I am left with two unknowns in my equation, that being the velocity and the mass. There is another equation I am not considering, I am sure of it.

 

[Doc Al]

W=F x d (work = force x distance)
W=PE + KE (work = potential energy + kinetic energy)

That's what you'll need. (There's no PE term in this problem.) But...

Therefore, I am left with two unknowns in my equation, that being the velocity and the mass. There is another equation I am not considering, I am sure of it.

You'll need to be given the mass to find the velocity. Ask your prof if he just forgot to specify it.

 

[Chestermiller]

Since he doesn't give the mass of the particle, you need to express your answers in terms of m.

Chet

 

[BvU]

Hello Randall, and welcome to PF.
Perhaps you want to make life easier for yourself and consider this as a horizontal movement. In other words, work is converted into kinetic energy. (Ergo no conservation as in your last eqn!). And if you aren't given the mass, you can't do better than express speed in terms of mass.

Well, we all agree...

 

[HalfLostAndSearching]

Your approach to part (c) is correct - the conservation of energy equation is the key to finding the speed of the particle at each slope change and x-intercept. However, in this case, you are correct in noting that the work done on and by the particle are not equal, so they cannot be set to zero in the conservation of energy equation.

To solve for the speed at each point, you will need to use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. This can be written as W = ΔKE. Using this equation, you can set the work done on the particle equal to the change in its kinetic energy and solve for the speed at each point.

For example, at the first slope change at x = 5.0m, the work done on the particle is equal to the area under the curve from x = 0.0m to x = 5.0m. This work is also equal to the change in the particle's kinetic energy from its initial speed (which is zero) to its speed at x = 5.0m. Using the work-energy theorem, you can set these two quantities equal to each other and solve for the speed at x = 5.0m.

You will need to repeat this process for each slope change and the x-intercept to find the speed at each point. Remember to also consider the direction of the work done on the particle - positive work will increase the kinetic energy, while negative work will decrease it.

I hope this helps you solve the problem!