6.2: Work and Kinetic Energy

In this section we will define work and kinetic energy. All of the descriptions below assume a world without friction. Friction is a force, and it can complicate a simple, straight-forward explanation. Unfortunately, our everyday experiences involve friction, and it can be difficult to eliminate it from our thinking about the world.

Work

In physics, the term "work" has a very specific and technical meaning that differs from its everyday use. It is not about effort or exertion, but rather about the transfer of energy. Work is the energy transferred to or from an object when a force acts on it, causing it to move over a distance. To be more precise, three conditions must be met for work to be done:

1. A force must be applied to an object. This force can be a push, a pull, gravity, friction, etc.
2. The object must move or be displaced. If you push on a wall with all your might, but the wall doesn't budge, you haven't done any work on the wall in the physics sense.
3. At least part of the force must be in the direction of the displacement. This is the most crucial part. If you are carrying a box horizontally, you are applying an upward force to support its weight. However, since the motion is horizontal (perpendicular to the force), no work is being done by your lifting force.

Figure \(\PageIndex{1}\): A schematic of an applied force doing work by displacing an object.

Key Aspects of Work

• Work is a scalar quantity: Unlike force and displacement which have direction, work does not. It is simply a value that represents a transfer of energy.
• Positive, Negative, and Zero Work:
  • Positive Work: Occurs when the force has a component in the same direction as the object's motion. This increases the object's energy. For example, pushing a box to make it speed up.
  • Negative Work: Occurs when the force has a component in the opposite direction of the object's motion. This removes energy from the object. For example, the force of friction acting on a sliding box is doing negative work, causing it to slow down.
  • Zero Work: Occurs when the force is perpendicular to the displacement, or when there is no displacement at all. For example, holding a heavy object still or carrying a briefcase at a constant speed on level ground.

In essence, work is the fundamental link between force and energy. It explains how forces change the energy of a system. The unit for work is the joule (J), which is the same unit used for energy. One joule is defined as the work done by a force of one newton acting over a distance of one meter. Mathematically, we describe work by the equation

\[W = Fd\]

where \(W\) is work, \(F\) is the applied force causing a displacement \(d\). It is important that we only consider displacement that is parallel to the applied force. In this text, we will only consider cases where the force and displacement are parallel.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion.

Simply put, if something is moving, it has kinetic energy. The amount of kinetic energy an object has depends on two key factors:

1. Its mass: A more massive object has more kinetic energy than a less massive object moving at the same speed. A bowling ball, for example, has far more kinetic energy than a tennis ball traveling at the same speed.
2. Its speed: This is the most significant factor. Kinetic energy is not just proportional to speed, it is proportional to the square of the speed. This means that if you double an object's speed, its kinetic energy increases by a factor of four (2²). If you triple the speed, the kinetic energy increases by a factor of nine (3²).

Written as an equation, kinetic energy is

\[KE = \frac{1}{2}mv^2\]

where \(KE\) is kinetic energy, \(m\) is the mass of the object that is moving with a speed \(v\).

Examples in Action:

• A moving car possesses a large amount of kinetic energy, which is why it can cause so much damage in a collision.
• A running person, a revolving planet, or a falling drop of rain all have kinetic energy.
• In the work-energy theorem, work done on an object can be understood as a transfer of energy that changes its kinetic energy. When you push a swing to make it move faster, you are doing work on it, and its kinetic energy increases.

In summary, kinetic energy is the energy of motion, and its value is a measure of how much work an object can do as a result of that motion. The faster and more massive an object is, the more kinetic energy it has.

The Work-Energy Theorem

The work-energy theorem is a fundamental principle in physics that connects the concepts of work and kinetic energy.

Imagine you have an object, like a box, that is sitting still on the floor. If you want to make it move, you have to push it. This act of pushing the box over a certain distance is what physicists call work.

Now, as you push the box, it starts to move faster and faster. The energy an object has because of its motion is called kinetic energy. The work-energy theorem simply states that the total amount of work you do on the box is equal to the change in its kinetic energy. Mathematically, this is

\[W = Fd = KE_{final}-KE_{initial}\]

\[W = \frac{1}{2}mv_{final}^2-\frac{1}{2}mv_{initial}^2\]

Let's reiterate work through a few simple scenarios:

• You do positive work: If you push the box in the direction it's already moving, you are adding energy to it. The box will speed up, and its kinetic energy will increase. The increase in kinetic energy is exactly equal to the work you performed.

• You do negative work: If the box is sliding, and you push against its direction of motion (like trying to slow it down), you are taking energy away from it. The box will slow down, and its kinetic energy will decrease. The decrease in kinetic energy is equal to the negative work you did.

• You do no work: If you push on a wall, you're applying a force, but the wall doesn't move. Since there is no displacement, no work is done on the wall, and its kinetic energy remains unchanged (it stays still).

The key takeaway is that work is a transfer of energy. When you do work on an object, you are either giving it kinetic energy or taking it away. The theorem is a powerful way to understand how forces and motion are related, without needing to go into the details of acceleration or time.

PhET Exploration: Work-Energy Theorem

In the PhET simulation below, click on the "Motion" tab. Check the boxes for Values, Masses, and Speed. In the simulation, there is no ruler for measuring displacements. However, the ground is paved with bricks. We can use the bricks as a measure of distance. Click the double arrows to the right and have the person push for two bricks of distance. Then, click on the person to stop pushing. Note the speed of the cart on the skateboard. Repeat this experiment for pushing four, six eight and ten blocks of distance. Each time record the final in a table like Table \(\PageIndex{2}\).

Distance Pushed (bricks)	Final Speed (m/s)
2	5.1
4	7.4
6	8.6
8	9.9
10	11.2

Table \(\PageIndex{2}\): A table of push distances and resulting speeds when the person pushes with 50 Newtons of force in the PhET simulaton.

In this example, we have a force that is 50 Newtons and a mass of 50 kg. The displacements are increasing. Therefore, the work \(W = Fd\) is increasing with distance, \(d\). As a result, we see the final kinetic energy increasing. (Your final speeds may vary a little from the ones in Table \(\PageIndex{2}\) due to the inexactness of measuring the number of bricks.) If we make a graph of final speed vs. distance (x vs. y), the result is Figure \(\PageIndex{3}\). Notice the graph is not a straight line.

Figure \(\PageIndex{3}\): A graph of final speed vs. distance for the PhET simulation.

In Figure \(\PageIndex{4}\) the final speed is graphed vs the square of the final speed.

Figure \(\PageIndex{4}\): A graph of final speed squared vs. distance for the PhET simulation.

This would be the same as the proportionality \(d\propto v^2\) or more explicitly

\[W = \frac{1}{2}mv^2\]

This graph appears to be linear as expected.