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Kinetic Energy
Kinetic energy is the energy of motion. Any object that is moving has kinetic energy, which depends on its mass and velocity. In AP Physics C, kinetic energy plays a central role in work-energy analysis and mechanical systems.
Definition and Formula
\[
K = \frac{1}{2}mv^2
\]
Where:
- K is the kinetic energy (in joules)
- m is the mass of the object (in kilograms)
- v is the velocity (in meters per second)
Kinetic energy is a scalar quantity, meaning it has magnitude but no direction. It always has a positive value (unless the object is at rest, in which case it’s zero).
Conceptual Understanding
- Doubling the velocity results in four times the kinetic energy, since velocity is squared.
- If two objects have the same kinetic energy but different masses, the one with less mass must be moving faster.
- Kinetic energy is frame-dependent: it changes with the observer’s frame of reference.
Calculus & Work-Energy
In AP Physics C, you’re expected to derive kinetic energy using calculus. If a net force \( F = ma \) acts on a particle over a displacement \( dx \), then:
\[
W = \int F \, dx = \int ma \, dx
\]
\[
= \int m \frac{dv}{dt} dx = \int m v \, dv = \frac{1}{2}mv^2
\]
This shows that the work done by a net force on an object is equal to the change in its kinetic energy:
\[
W_{\text{net}} = \Delta K
\]
Example
A 2 kg object moves at 3 m/s. What is its kinetic energy?
Solution:
\[
K = \frac{1}{2}(2)(3)^2 = 9 \, \text{J}
\]
Interactive Kinetic Energy Mini-Lab