7.2 Kinetic Energy and the Work-Energy Theorem

Kinetic energy and work are closely linked concepts in physics. The work-energy theorem gives you a powerful shortcut: instead of tracking forces and acceleration over time, you can directly connect the work done on an object to how its speed changes. This section covers how to calculate kinetic energy, apply the work-energy theorem, and understand how forces transfer energy to objects in motion.

Kinetic Energy and Work-Energy Theorem

Work-energy theorem calculations

Kinetic energy is the energy an object has because it's moving. The faster it moves or the more massive it is, the more kinetic energy it carries.

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

• $m$ is the mass of the object (in kg)
• $v$ is the velocity of the object (in m/s)

For example, a 2 kg ball moving at 3 m/s has a kinetic energy of $\frac{1}{2}(2)(3)^2 = 9 \text{ J}$. Notice that kinetic energy depends on velocity squared, so doubling the speed quadruples the kinetic energy.

Work is the energy transferred to an object by a force acting over a displacement.

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

• $F$ is the magnitude of the applied force (in N)
• $d$ is the displacement (in m)
• $\theta$ is the angle between the force and the direction of displacement

If you push a box 5 m with a force of 20 N at 30° to the horizontal, the work done is $W = (20)(5)\cos 30° = 100 \times 0.866 \approx 86.6 \text{ J}$. Only the component of force along the displacement does work.

The work-energy theorem ties these together: the net work done on an object equals the change in its kinetic energy.

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

This means you don't need to find acceleration or use kinematics. If you know the net work, you know how the kinetic energy changed.

Kinetic energy in energy transfer

When work is done on an object, energy is transferred to or from it:

• Positive work increases kinetic energy (the object speeds up). This happens when the force has a component in the direction of motion.
• Negative work decreases kinetic energy (the object slows down). This happens when the force opposes the motion, like friction or braking.
• Zero work occurs when the force is perpendicular to the displacement ($\cos 90° = 0$), such as the normal force on a box sliding across a flat floor.

Conservation of energy applies when only conservative forces (like gravity) act on a system. In that case, the total mechanical energy stays constant:

$\Delta KE + \Delta PE = 0$

This can be rearranged to $\Delta KE = -\Delta PE$. A roller coaster at the top of a hill has high potential energy and low kinetic energy. As it drops, potential energy converts to kinetic energy, and the coaster speeds up. At the bottom, the situation reverses: high kinetic energy, low potential energy.

Power measures how quickly work is done or energy is transferred:

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

• $P$ is power (in watts, W)
• $\Delta t$ is the time interval (in s)

A 100 W light bulb converts 100 J of energy every second. A more powerful engine doesn't necessarily do more total work; it does the same work in less time.

Forces in work-energy problems

The net force on an object determines whether its kinetic energy changes. Here's how to think through a work-energy problem:

1. Identify all forces acting on the object (gravity, applied force, friction, normal force, etc.).
2. Find the net force by adding all forces as vectors. If the net force is zero, the velocity stays constant and kinetic energy doesn't change.
3. Calculate the net work using $W_{net} = F_{net} \cdot d \cdot \cos\theta$.
4. Apply the work-energy theorem to find the change in kinetic energy: $W_{net} = \Delta KE$.

For example, a 5 kg box is pushed 3 m by a 10 N force at 60° to the horizontal. The work done by that force is $W = (10)(3)\cos 60° = 30 \times 0.5 = 15 \text{ J}$. If this is the only force doing work (normal force and gravity are perpendicular to the displacement on a flat surface), then $\Delta KE = 15 \text{ J}$.

When non-conservative forces like friction are present, mechanical energy is not conserved. Friction converts kinetic energy into thermal energy (heat), so the total mechanical energy of the system decreases. A pendulum in a vacuum swings forever, but a real pendulum gradually loses energy to air resistance and friction at the pivot.

Energy, Momentum, and Motion

Kinetic energy and momentum both describe a moving object, but they behave differently:

• Momentum: $p = mv$ (scales linearly with velocity, is a vector)
• Kinetic energy: $KE = \frac{1}{2}mv^2$ (scales with velocity squared, is a scalar)

Because kinetic energy depends on $v^2$, a small increase in speed at high velocity adds much more kinetic energy than the same increase at low velocity. A car going from 10 m/s to 20 m/s gains far more kinetic energy than one going from 0 to 10 m/s, even though both increase speed by 10 m/s.

Energy conversion happens constantly in physical systems. In a hydroelectric dam, the gravitational potential energy of water at height converts to kinetic energy as the water falls, which then spins turbines to generate electrical energy. At each stage, the total energy is conserved, though some is always lost to heat through friction and resistance.