Objects can possess energy in many forms but this section is only going to consider the energy from the translational motion of the system. This motional energy is called kinetic energy ($K = \frac{1}{2} m v^2$) and notice it is a scalar quantity. It is true that velocity is a vector quantity but when you square it, all direction is lost. Objects can also exchange energy with other objects through the concept of Work ($W$). The Work-Kinetic Energy theorem relates the change in kinetic energy of the system to the work performed on the system.

Check out this song about all the forms of energy.

https://www.youtube.com/watch?v=mHw34osLy4Q

Pre-lecture Study Resources

Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.

BoxSand Introduction

Work and Energy| Work and Kinetic Energy Theorem

Consider the motion in 1D and the kinematics equation, $v_{fx}^2 = v_{ix}^2 + 2a_x \Delta x$, and Newton's 2nd Law with constant acceleration in one dimension, $\sum F_x = m a_x$. A simple substitution, and minor rearrangements, of the acceleration from the 2nd law into the kinematics equation yields:

$\frac{1}{2}m v_{fx}^2 - \frac{1}{2}m v_{ix}^2 = \sum F_x \Delta x$

It appears that forces acting over a distance changes the velocity of objects according to the equation above. As it turns out, this manipulation of equations will be of great benefit in analyzing the motion of objects, so we define new terms for the expressions shown in the above equation.

Linear Kinetic Energy (K): $\frac{1}{2}m v^2$.

Work (W): $F_x \Delta x$.

Using our newly defined terms, we now can write what we call, the Work-Kinetic Energy theorem as:

$K_f - K_i = \sum W$

This expression lets you see the heart of the Work-Kinetic Energy theorem, namely that a system's change in energy is due to work from external forces performed on the system. The theorem is often rearranged so that it can be interpreted as: take the initial kinetic energy of the system and add the net work acting on the system and this is equal to the final kinetic energy of the system.

$K_i + \sum W = K_f$

**Note: K and W are scalar quantities! The above arguments are shown in one dimension. Continue reading further in this section to learn how to generalize if our forces, displacements, or velocities are in more than one dimension.

Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

BoxSand Videos

OpenStax Reading


OpenStax Section 7.1  |  Work: The Scientific Definition

Openstax College Textbook Icon

OpenStax Section 7.2  |  Kinetic Energy and the Work-Kinetic Energy Theorem

Openstax College Textbook Icon

Fundamental examples


1. An object initially has a kinetic energy of $7 \, J$. A force is then applied on the object as described in the graph below. What is the final kinetic energy of this object after the force acts on the object over $3 $ meters?

 

This is a graph of the net force in newton over position. the graph initially starts at four newtons and decreases to negative two seconds over 3 meters.

 

2. A box of mass $4 \, kg$ initially at rest sits on top of a frictionless table. A constant force of magnitude $3 \, N$ is then applied to the box as shown in the image below. What is the change in kinetic energy of the box if the force is applied over a distance of $5 \, m$?

 

This is a representation of a box on top of a frictionless table with some force p applied to the left that pushes the box to the right

 

3. A box of mass $4 \, kg$ initially at rest sits on top of a frictionless table. A constant force of magnitude $3 \, N$ is then applied to the box as shown in the image below. What is the change in kinetic energy of the box if the force is applied over a distance of $5 \, m$?

 

This is a representation of a box on top of a frictionless table with some force p applied to the top left corner of the box that applies a force downward and to the right at some angle theta to the horizontal

 

CLICK HERE for solutions.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

Clicker Questions Icon

 

Post-Lecture Study Resources

Use the supplemental resources below to support your post-lecture study.

Practice Problems

Recommended example practice problems

BoxSand's Quantitative Practice Problems

BoxSand's Multiple Select Problems

BoxSand's Conceptual Problems 

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

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Additional Boxsand Study Resources

Additional BoxSand Study Resources

Learning Objectives

Summary

Students are introduced to the concept of kinetic energy and how work is a mechanism for systems to exchange energy. The goal is construct the work-energy theorem to analyze systems which requires knowledge of the dot product.

Atomistic Goals

Students will be able to...

  1. Find the kinetic energy for a system.
  2. Identify which quantities are scalars and which are vectors. 
  3. Define the work-energy theorem and explain how work is a mechanism to transfer energy between systems.
  4. (UPMF) Differentiate between internal and external work.
  5. Demonstrate that energy involves forces applied over distances and momentum involves forces applied over time.
  6. (UPMF) Find the area under a net force vs displacement curve and set that equal to the work on the system.  
  7. Identify that work is a dot product between the force and the change in position.
  8. Determine whether work is positive or negative based on forces and displacement.
  9. Determine whether (net) work is positive or negative based on the change in kinetic energy.
  10. Identify that a dot product depends on the magnitude of the two vectors and the smallest angle between them.  
  11. Explain how a dot product shows how much one vector points in the same direction as another using the projection method.
  12. (UPMF) Use a vector operation diagram to visualize the dot product between two vectors.
  13. Calculate work by applying the dot product in the mathematical form via magnitudes and smallest angles.
  14. Calculate work by applying the dot product in the mathematical form of summation of products of components.
  15. Construct the work-energy theorem in the mathematical representation to analyze physical systems.

 

Equations, definitions, and notation icon Concept Map Icon
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YouTube Videos

Work-Energy theorem and pole vaulting.

https://www.youtube.com/watch?v=pmOXi-My6ZI

Simulations


Check out this interactive simulation on kinetic energy. Note that the external work in this case is due to friction if it is turned on.

Phet Interactive Simulations Icon

 

Build your own roller coaster and test to see if it has enough initial energy to get through the entire track.

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For additional simulations on this subject, visit the simulations repository.

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Demos


Simple work enegry theorem demonstration with a toy car and motion detector,

https://www.youtube.com/watch?v=lBdHx3JX9DA

 

For additional demos involving this subject, visit the demo repository

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History

Physics Fun


Oh no, we haven't been able to post any fun stuff for this topic yet. If you have any fun physics videos or webpages for this topic, send them to the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

Other Resources


This link introduces the concept of kinetic energy.

The Physics Classroom Icon

This link provides an introduction to the concept of work.

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Other Resources

This link will take you to the repository of other content related resources.

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Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions


  1.  
  2.  

Checklist

1. Read and re-read the whole problem carefully.

2. Visualize the scenario. Mentally try to understand what the object(s) are doing and what forces are acting on them.

3. Identify the system you wish to analyze; This may include one or more objects if coupled to one another.

4. Write down all of your knowns and unknowns.

5. Identify all the forces acting on your object(s). Include a FBD to help keep track of all the forces.

a. If a force acting on an object is not constant, (i.e. it depends on position), then consult a force vs displacement graph to calculate the work.

6. Determine the displacement of the object.

7. Calculate the work done by each force.

a. For forces that are not parallel or perpendicular, draw a vector operation to help determine the angle between the force and displacement.

b. Double check you have the right sign of work.

8. Use the work-kinetic energy theorem $KE_i + \sum W = KE_f$ to solve for the relevant quantities.

9. Evaluate your answer; make sure units are correct and the results are within reason.

Misconceptions & Mistakes

Kinetic Energy

  • Remember, kinetic energy is not related to size, only mass and velocity.
  • Kinetic energy and work are both scalar quantities; do not attempt to split them up into components like vectors.
  • Kinetic energy occurs whenever an object has a velocity, no matter the reason for the motion.
  • You can't always assume that higher velocity (or larger mass) means larger kinetic energy. You must look at both the mass and the velocity of the object.
  • A big misconception is that doubling the velocity doubles the kinetic energy. When you actually look at the equation: $KE=\frac{1}{2}mv^2$ you can see that the velocity is squared thus if you double the velocity you actually increase the kinetic energy by that squared factor.

Work

  • Just because a large force is applied does not necessarily mean work was done. The object must move for to have been done.
  • Work can be negative or positive- depending on whether the system is doing work (-) or whether work is being done on the system (+).
  • Objects do not possess work; work is a mechanism for interacting objects to exchange kinetic energy.
  • Think about whether it is being done on the system or by the system and use that to double check the sign of the work.
  • If the force acting over a distance is not constant, you can not just use the maximum value of the force. You must consult a force vs distance graph and use the area under the curve to estimate the work done.

Pro Tips

  • If given information about forces and displacements, start to think about work-kinetic energy.
  • For forces that are not parallel or perpendicular to displacements, draw a vector operation to ensure you are using the correct angle for the dot product between the two vectors.

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways. 

Physical


Physical Representations describes the physical phenomena of the situation in a visual way.

 

A block is launched up a frictionless ramp with initial velocity $\vec{v_{i}}$.

This is an image of a block on a ramp that has some horizontal force applied from the left that pushes the block up the ramp to the right. the block was moving at some initial velocity up the ramp and at a final velocity at an angle at the top of the ramp. The ramp is at some angle theta from the vertical at the top of the ramp and is at some height h in the y axis. y equals zero at the bottom of the ramp.

 

Mathematical


Mathematical Representation uses equation(s) to describe and analyze the situation.

$ \sum K_{i} + \sum W_{external} = \sum K_{f}$

$ K = \frac{1}{2} mv^{2}$

$W = \bar{\vec{F}_{ext}} \cdot \Delta \vec{r}$ 

A representation with the words linear kinetic energy (k) on the top. There is an equation that shows that the linear kinetic energy is equal to one-half the mass multiplied by the velocity squared. This is also written in words below.

A representation with the words work (w) on the top. There is an equation that shows that the work is equal to the dot product of the average external force and change in position, which is also equal to the product of the magnitudes of the average external force and change in position, multiplied by the smallest angle between the external force vector and the change in position vector. This is also written in words below

A representation with the words work-energy theorem on the top. There is an equation that shows that a system change in energy is the sum of the final kinetic energy minus the sum of the initial kinetic energy, which is equal to the work done on the system by the environment. This is also written in words below

 

Graphical


Graphical Representation describes the situation through use of plots and graphs.

This is a graph of force over position graph where the force starts at some value and decreases at some position and is constant until it reaches some final position. .

Descriptive


Descriptive Representation describes the physical phenomena with words and annotations.

A block is launched up a frictionless ramp, as shown in physical representation above, with an initial $\vec{v_{i}}$. The block travels up the ramp and continues across the level section. Once the block has been launched up the ramp, force due to gravity and the normal force will act upon the block. The force due to gravity will slow the box down, so gravity will do non-zero work on the block, while the normal force will do zero work on the block. If friction was present, then friction would also be doing work on the block and decelerate the block.

 

Experimental


Experimental Representation examines a physical phenomena through observations and data measurement.