Though energy is always measured in joules, and though it can always be defined as a capacity to do work, energy manifests itself in a variety of different forms. These various forms pop up all over SAT II Physics, and we will look at some additional forms of energy when we discuss electromagnetism, relativity, and a number of other specialized topics. For now, we will focus on the kinds of energy you’ll find in mechanics problems.
Kinetic Energy
Kinetic energy is the energy a body in motion has by virtue of its motion. We define energy as the capacity to do work, and a body in motion is able to use its motion to do work. For instance, a cue ball on a pool table can use its motion to do work on the eight ball. When the cue ball strikes the eight ball, the cue ball comes to a stop and the eight ball starts moving. This occurs because the cue ball’s kinetic energy has been transferred to the eight ball.
There are many types of kinetic energy, including vibrational, translational, and rotational. Translational kinetic energy, the main type, is the energy of a particle moving in space and is defined in terms of the particle’s mass, m, and velocity, v:
For instance, a cue ball of mass 0.5 kg moving at a velocity of 2 m/s has a kinetic energy of 1/2 (0.5 kg)(2 m/s)2 = 1 J.
The Work-Energy Theorem
If you recall, work is a measure of the transfer of energy. An object that has a certain amount of work done on it has that amount of energy transferred to it. This energy moves the object over a certain distance with a certain force; in other words, it is kinetic energy. This handy little fact is expressed in the work-energy theorem, which states that the net work done on an object is equal to the object’s change in kinetic energy:
For example, say you apply a force to a particle, causing it to accelerate. This force does positive work on the particle and increases its kinetic energy. Conversely, say you apply a force to decelerate a particle. This force does negative work on the particle and decreases its kinetic energy. If you know the forces acting on an object, the work-energy theorem provides a convenient way to calculate the velocity of a particle.
Example
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A hockey puck of mass 1 kg slides across the ice with an initial velocity of 10 m/s. There is a 1 N force of friction acting against the puck. What is the puck’s velocity after it has glided 32 m along the ice? | |
If we know the puck’s kinetic energy after it has glided 32 m, we can calculate its velocity. To determine its kinetic energy at that point, we need to know its initial kinetic energy, and how much that kinetic energy changes as the puck glides across the ice.
First, let’s determine the initial kinetic energy of the puck. We know the puck’s initial mass and initial velocity, so we just need to plug these numbers into the equation for kinetic energy:
The friction between the puck and the ice decelerates the puck. The amount of work the ice does on the puck, which is the product of the force of friction and the puck’s displacement, is negative.
The work done on the puck decreases its kinetic energy, so after it has glided 32 m, the kinetic energy of the puck is 50 – 32 = 18 J. Now that we know the final kinetic energy of the puck, we can calculate its final velocity by once more plugging numbers into the formula for kinetic energy:
