# Mechanics

calculus-based physics for scientists and engineers

## Work and Kinetic Energy in One Dimension

### Kinetic Energy

Energy can be very complicated ... unless you have only particles! In our particle model, a particle exists only at a point and has no internal structure. For a particle, the only energy is that of motion: Kinetic Energy. This will come as a shock, no doubt, but the more energy something has, the faster it is going.

The kinetic energy of any object is given by $$K=\frac{1}{2}mv^2$$ where $$m$$ is the mass and $$v$$ is the speed. This is one of those equations you just have to memorize. But let's face it, it isn't that hard to memorize. Another fact to always have handy is that kinetic energy is proportional to speed squared. Just this fact, and proportional reasoning, can solve many problems.

Kinetic Energy is a scalar. Even though we are only in 1D now, the above holds in two and three dimensions as well. Since your velocity (speed) depends on your coordinate system, so does your kinetic energy! It is not an objective quantity, like mass. We explore that (and more!) in the video below.

### Work - Kinetic Energy Theorem

In general, work can change the energy of a system in complicated ways. However, a particle is not complicated. A particle is a point without structure. A particle can only have kinetic energy, or energy of motion. Thus, the net work done on a particle changes its kinetic energy.

The work-kinetic energy theorem (W-KE theorem) says exactly what you expect: for a particle, the net work is the change in kinetic energy. That seems simple enough, but here is a list of errors you will be making if you are not careful:

1. This only works for a particle! I cannot stress this enough. If you have a system of particles, this is not true. Other things can happen, like the potential energy of the system can change.
2. Only the net work changes the kinetic energy. Calculating the work from one force is not sufficient. You have to calculate the net work.
3. Work is the change in kinetic energy, not the kinetic energy itself.
4. Work can be negative, but kinetic energy cannot. Positive work increases the kinetic energy of the particle. Negative work decreases the kinetic energy. You cannot decrease the kinetic energy below zero.

### Solution

1. A 1900 kg elevator accelerates upward at a constant acceleration of 1.45 m/s2 for 12 m, starting from rest. Use work to find the final speed of the system.

### Solution

2. You throw a ball up with speed, $$v$$ and it reaches a height, $$h$$. Using work-KE theorem, at what fraction of the total height was the speed $$v/2$$?