Glossary
Air Resistance
A common type of resistive force caused by an object moving through air. It is often modeled as proportional to velocity or velocity squared.
Example:
A cyclist experiences significant air resistance when riding into a strong headwind, making it harder to maintain speed.
Asymptotes
Lines that a curve approaches as it heads towards infinity. In physics, they represent the limiting values that physical quantities, like velocity or acceleration, approach over time.
Example:
The velocity of a falling object with air resistance will approach a horizontal asymptote, which represents its terminal velocity.
Differential Equations
Equations that involve an unknown function and its derivatives. In mechanics, they are used to describe how quantities like velocity and position change over time due to forces.
Example:
Modeling the decay of a radioactive substance or the motion of a pendulum often requires solving differential equations.
Exponential Functions
Functions where the independent variable appears in the exponent. In resistive force problems, velocity, position, and acceleration often follow exponential decay or growth patterns.
Example:
The charging of a capacitor in an RC circuit is described by an exponential function, showing how the voltage approaches its maximum value over time.
Newton's Second Law
States that the net force acting on an object is equal to the product of its mass and acceleration (F_net = ma). It is fundamental for setting up equations of motion.
Example:
To calculate the acceleration of a rocket, engineers apply Newton's Second Law, considering the thrust and gravitational forces acting on it.
Resistive Forces
Forces that always oppose the motion of an object, slowing it down. Their magnitude typically increases with the object's speed.
Example:
When a car brakes, the friction between the tires and the road acts as a resistive force, bringing the vehicle to a stop.
Separation of Variables
A technique used to solve certain types of differential equations by rearranging the equation so that each variable is on a different side, allowing for independent integration.
Example:
When solving for the velocity of an object falling with air resistance, separation of variables allows us to integrate the velocity terms separately from the time terms.
Terminal Velocity
The constant maximum speed an object reaches when the resistive force balancing a constant applied force (like gravity) results in zero net acceleration.
Example:
A skydiver reaches terminal velocity when the upward air resistance perfectly balances the downward force of gravity, causing them to fall at a constant speed.
Time Constant (τ)
A characteristic time scale for systems exhibiting exponential behavior, defined as m/k for linear resistive forces. It indicates how quickly a system responds or approaches its steady state.
Example:
A larger time constant for a car's braking system means it takes longer for the car to come to a complete stop after the brakes are applied.