6.2 Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy definition and formula

Kinetic energy (KE) is the energy an object has because it's moving. It's a scalar quantity, meaning it has magnitude but no direction, and it's measured in joules (J).

The formula is:

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

where $m$ is the object's mass (in kg) and $v$ is its velocity (in m/s).

The two factors that determine kinetic energy behave very differently:

• Mass has a linear relationship with KE. Double the mass, and KE doubles.
• Velocity has a squared relationship with KE. Double the velocity, and KE quadruples.

That squared relationship is why velocity matters so much more than mass in practice. A 0.01 kg bullet traveling at 700 m/s carries far more kinetic energy than a 0.15 kg baseball thrown at 40 m/s, even though the baseball is 15 times heavier. It's also why highway car crashes are so much more dangerous than low-speed fender benders.

Work-Energy Theorem

Work-energy theorem application

The work-energy theorem states that the net work done on an object equals its change in kinetic energy:

$W_{net} = \Delta KE = KE_f - KE_i$

This means you don't need to track every force and acceleration to figure out how fast something ends up moving. If you know the total work done on it, you know how its kinetic energy changed.

To apply the theorem, follow these steps:

1. Calculate the net work done on the object (sum of work from all forces).
2. Determine the initial kinetic energy using $KE_i = \frac{1}{2}mv_i^2$.
3. Solve for the final kinetic energy: $KE_f = KE_i + W_{net}$.
4. If you need the final speed, rearrange to get $v_f = \sqrt{\frac{2 \cdot KE_f}{m}}$.

The theorem is really an alternative way of expressing Newton's Second Law ($F = ma$). Instead of working with forces and accelerations directly, you're working with energy. This is especially useful when forces vary over a distance, like a spring pushing a block, because calculating work can be simpler than tracking changing acceleration.

Kinetic energy changes from work

The change in kinetic energy is:

$\Delta KE = KE_f - KE_i$

And work done by a constant force is:

$W = F \cdot d \cdot \cos\theta$

where $F$ is the force magnitude, $d$ is the displacement, and $\theta$ is the angle between the force and the direction of motion.

The sign of the work tells you what happens to the object's motion:

• Positive work increases kinetic energy. A force applied in the direction of motion speeds the object up (e.g., pushing a stalled car forward).
• Negative work decreases kinetic energy. A force opposing the motion slows the object down (e.g., brake pads applying friction to a bicycle wheel).
• Zero work means no change in kinetic energy. This happens when the force is perpendicular to the motion (like gravity acting on an object moving horizontally) or when there's no displacement (holding a heavy book stationary).

Relationship of energy forms

The conservation of energy principle says that total energy in a closed system stays constant. Energy isn't created or destroyed; it just changes form.

Mechanical energy is the sum of kinetic energy and potential energy in a system. The two main types of potential energy you'll encounter in this course are:

• Gravitational potential energy, which depends on an object's height above a reference point.
• Elastic potential energy, which is stored in stretched or compressed springs.

Energy transforms back and forth between these forms. A pendulum at the top of its swing has maximum potential energy and zero kinetic energy. At the bottom of its swing, that potential energy has converted into kinetic energy, and the pendulum moves at its fastest.

In real systems, non-conservative forces like friction and air resistance convert mechanical energy into thermal energy (heat). That's why a ball bouncing on the floor eventually stops: each bounce loses some mechanical energy to heat.

Power measures how quickly work is done or energy is transferred. It's calculated as:

$P = \frac{W}{t}$

or equivalently, for a constant force applied along the direction of motion:

$P = F \cdot v$

Power is measured in watts (W), where 1 watt = 1 joule per second. A 100 W light bulb converts 100 joules of electrical energy every second. Car engines are often rated in horsepower, where 1 hp ≈ 746 W.